Almost minimizers for a sublinear system with free boundary

Abstract

We study vector-valued almost minimizers of the energy functional

$$\begin{aligned} \int _D\left( |\nabla \textbf{u}|^2+\frac{2}{1+q}\left( \lambda _+(x)|\textbf{u}^+|^{q+1}+\lambda _-(x)|\textbf{u}^-|^{q+1}\right) \right) dx,\quad 0<q<1. \end{aligned}$$

For Hölder continuous coefficients $\lambda _\pm (x)>0$, we take the epiperimetric inequality approach and prove the regularity for both almost minimizers and the set of “regular" free boundary points.

1 Introduction

1.1 A sublinear elliptic system

Let D be a domain in ${{\mathbb {R}}}^n$, $n\ge 2$, and $\textbf{ g}:\partial D\rightarrow {{\mathbb {R}}}^m$, $m\ge 1$, be a given function (the boundary value). We also let $\lambda _\pm (x)$ be $\alpha$-Hölder continuous functions satisfying $0<\lambda _0\le \lambda _\pm (x)\le \lambda _1<\infty$ for some positive constants $\lambda _0$, $\lambda _1$. Then, for $0<q<1$ and $F(x,\textbf{v})=\frac{1}{1+q}\left( \lambda _+(x)|\textbf{v}^+|^{q+1}+\lambda _-(x)|\textbf{v}^-|^{q+1}\right)$, consider the minimizer $\textbf{u}$ of the energy functional

$$\begin{aligned} \int _D\left( |\nabla \textbf{v}|^2+2F(x,\textbf{v})\right) dx, \end{aligned}$$

(1.1)

among all functions $\textbf{v}\in W^{1,2}(D;{{\mathbb {R}}}^m)$ with $\textbf{v}=\textbf{ g}$ on $\partial D$. It is well-known that there exists a unique minimizer $\textbf{u}$ and it solves a sublinear system

$$\begin{aligned} \Delta \textbf{u}=\lambda _+(x)|\textbf{u}^+|^{q-1}\textbf{u}^+-\lambda _-(x)|\textbf{u}^-|^{q-1}\textbf{u}^-. \end{aligned}$$

The regularity of both the solution $\textbf{u}$ and its free boundary $\Gamma (\textbf{u}):=\partial \{x:\,|\textbf{u}(x)|>0\}$ was studied in [4] or in the scalar case (when $m=1$) in [3].

1.2 Almost minimizers

In this paper we consider almost minimizers of the functional (1.1).

To introduce the definition of almost minimizers, we let $\omega :(0,r_0)\longmapsto [0,\infty )$, $r_0>0$, be a gauge function, which is a nondecreasing function with $\omega (0+)=0$.

Definition 1

(Almost minimizers) Let $0<r_0<1$ be a constant and $\omega (r)$ be a gauge function. We say that a function $\textbf{u}\in W^{1,2}(B_1;{{\mathbb {R}}}^m)$ is an almost minimizer for the functional $\int \left( |\nabla \textbf{u}|^2+2F(x,\textbf{u})\right) \,dx$ in a domain D, with gauge function $\omega (r)$, if for any ball $B_r(x_0)\Subset D$ with $0<r<r_0$, we have

$$\begin{aligned} \int _{B_r(x_0)}\left( |\nabla \textbf{u}|^2+2F(x,\textbf{u})\right) dx\le (1+\omega (r))\int _{B_r(x_0)}\left( |\nabla \textbf{v}|^2+2F(x,\textbf{v})\right) dx, \end{aligned}$$

(1.2)

for any competitor function $\textbf{v}\in \textbf{u}+W^{1,2}_0(B_r(x_0);{{\mathbb {R}}}^m)$.

In fact, we can observe that for x, $x_0\in D$,

$$\begin{aligned} 1-C|x-x_0|^\alpha \le \frac{\lambda _\pm (x)}{\lambda _\pm (x_0)}\le 1+C|x-x_0|^\alpha , \end{aligned}$$

(1.3)

with a constant C depending only on $\lambda _0$ and $\Vert \lambda _\pm \Vert _{C^{0,\alpha }(D)}$. Using this, we can rewrite (1.2) in the form with frozen coefficients

$$\begin{aligned} \int _{B_r(x_0)}\left( |\nabla \textbf{u}|^2+2F(x_0,\textbf{u})\right) dx\le (1+\widetilde{\omega }(r))\int _{B_r(x_0)}\left( |\nabla \textbf{v}|^2+2F(x_0,\textbf{v})\right) dx, \end{aligned}$$

(1.4)

where

$$\begin{aligned} F(x_0,\textbf{u})&=\frac{1}{q+1}\left( \lambda _+(x_0)|\textbf{u}^+|^{q+1}+\lambda _-(x_0)|\textbf{u}^-|^{q+1}\right) ,\\ \widetilde{\omega }(r)&=C(\omega (r)+r^\alpha ). \end{aligned}$$

This implies that almost minimizers of (1.1) with Hölder coefficients $\lambda _\pm$ are almost minimizers with frozen coefficients (1.4).

An example of an almost minimizer can be found in Appendix A. Almost minimizers for the case $q=0$ and $\lambda _\pm =1$ were studied by the authors in [2], where the regularity of both the almost minimizers and the regular part of the free boundary has been proved.

In this paper we aim to extend the results in [4] from solutions to almost minimizers and those in [2] from the case $q=0$ to $0<q<1$.

1.3 Main results

Due to the technical nature of the problem, we assume that the gauge function $\omega (r)=r^\alpha$ for $0<\alpha <2$, D is the unit ball $B_1$, and the constant $r_0=1$ in Definition 1.

In addition, to simplify tracking all constants, we take $M>2$ such that

$$\begin{aligned} \Vert \lambda _\pm \Vert _{C^{0,\alpha }(B_1)}\le M,\qquad \frac{1}{\lambda _0},\,\lambda _1\le M,\qquad \omega (r),\,\widetilde{\omega }(r)\le Mr^\alpha . \end{aligned}$$

(1.5)

Now, we state our main results.

Theorem 1

(Regularity of almost minimizers) Let $\textbf{u}\in W^{1,2}(B_1;{{\mathbb {R}}}^m)$ be an almost minimizer in $B_1$. Then $\textbf{u}\in C^{1,\alpha /2}(B_1)$. Moreover, for any $K\Subset B_1$,

$$\begin{aligned} \Vert \textbf{u}\Vert _{C^{1,\alpha /2}(K;{{\mathbb {R}}}^m)}\le C(n,\alpha ,M,K)(E(\textbf{u},1)^{1/2}+1), \end{aligned}$$

(1.6)

where $E(\textbf{u},1)=\int _{B_1}\left( |\nabla \textbf{u}|^2+|\textbf{u}|^{q+1}\right)$.

This regularity result is rather immediate in the case of minimizers (or solutions), since their $W^{2,p}$-regularity for any $p<\infty$ simply follows from the elliptic theory with a bootstrapping. This is inapplicable to almost minimizers, as they do not satisfy a partial differential equations. Instead, we follow the approach in [2] by first deriving growth estimates for almost minimizers and then using Morrey and Campanato space embedding theorems.

To investigate the free boundary, for $\kappa :=\frac{2}{1-q}>2$ we define a subset $\Gamma ^\kappa (\textbf{u})$ of the free boundary $\Gamma (\textbf{u})=\partial \{|\textbf{u}|>0\}$ as

$$\begin{aligned} \Gamma ^\kappa (\textbf{u}):=\{x_0\in \Gamma (\textbf{u})\,:\, \textbf{u}(x)=O(|x-x_0|^{\xi }) \ \hbox {for some } \lfloor \kappa \rfloor<\xi <\kappa \}.\end{aligned}$$

(1.7)

Here, the big O is not necessarily uniform on $\textbf{u}$ and $x_0$, and $\lfloor s\rfloor$ is the greatest integer less than s, i.e., $s-1\le \lfloor s\rfloor < s$.

In the study of solutions in [4], the authors considered a subset of the free boundary, which essentially consists of points where all derivatives of order less than $\lfloor \kappa \rfloor$ are zero. This subset is slightly larger than $\Gamma ^\kappa (\textbf{u})$ in (1.7).

Theorem 2

(Optimal growth estimate) Let $\textbf{u}$ be as in Theorem 1. Then there are constants $C>0$ and $r_0>0$, depending only on n, $\alpha$, M, $\kappa$, $E(\textbf{u},1)$, such that

$$\begin{aligned} \int _{B_r(x_0)}\left( |\nabla \textbf{u}|^2+|\textbf{u}|^{1+q}\right) \le Cr^{n+2\kappa -2} \end{aligned}$$

for $x_0\in \Gamma ^\kappa (\textbf{u})\cap B_{1/2}$ and $0<r<r_0$.

The proof is inspired by the ones for minimizers in [4] and for the case $q=0$ in [2]. However, in our case concerning almost minimizers with $\kappa >2$, several new technical difficulties arise and the proof is much more complicated, as we have to improve the previous techniques by using approximation by harmonic polynomials and limiting argument.

One implication of Theorem 2 is the existence of $\kappa$-homogeneous blowups (Theorem 8). This allows to consider a subset of $\Gamma ^\kappa (\textbf{u})$, the so-called “regular" set, using a class of half-space solutions

$$\begin{aligned} {{\mathbb {H}}}_{x_0}&:= \{ x\mapsto \beta _{x_0}\max (x\cdot \nu ,0)^\kappa \textbf{e} : \nu \in {{\mathbb {R}}}^n \text { and } \textbf{e}\in {{\mathbb {R}}}^m \text { are unit vectors}\},\quad x_0\in B_1. \end{aligned}$$

Definition 2

(Regular free boundary points) We say that a point $x_0\in \Gamma ^\kappa (\textbf{u})$ is a regular free boundary point if at least one homogeneous blowup of $\textbf{u}$ at $x_0$ belongs to ${{\mathbb {H}}}_{x_0}$. We denote by ${\mathcal {R}}_\textbf{u}$ the set of all regular free boundary points in $\Gamma (\textbf{u})$ and call it the regular set.

The following is our central result concerning the regularity of the free boundary.

Theorem 3

(Regularity of the regular set) ${\mathcal {R}}_\textbf{u}$ is a relatively open subset of the free boundary $\Gamma (\textbf{u})$ and locally a $C^{1,\gamma }$-manifold for some $\gamma =\gamma (n,\alpha , q,\eta )>0$, where $\eta$ is the constant in Theorem 7.

The proof is based on the use of the epiperimetric inequality from [4] and follows the general approach in [2, 6]: The combination of the monotonicity of Weiss-type energy functional (Theorem 5) and the epiperimetric inequality, together with Theorem 2, establishes the geometric decay rate for the Weiss functional. This, in turn, provides us with the rate of convergence of proper rescalings to a blowup, ultimately implying the regularity of ${\mathcal {R}}_\textbf{u}$.

1.4 Plan of the paper

The plan of the paper is as follows.

In Sect. 2 we study the regularity properties of almost minimizers. We prove their almost Lipschitz regularity (Theorem 4) and exploit it to infer the $C^{1,\alpha /2}$-regularity (Theorem 1).

In Sect. 3 we establish the Weiss-type monotonicity formula (Theorem 5), which will play a significant role in the analysis of the free boundary.

Section 4 is dedicated to providing the proof of the optimal growth estimates in Theorem 2 above.

Section 5 is devoted to proving the non-degeneracy result of almost minimizers, following the line of [2].

In Sect. 6 we discuss the homogeneous blowup of almost minimizers at free boundary points, including its existence and properties. In addition, we estimate a decay rate of the Weiss energy, with the help of the epiperimetric inequality.

In Sect. 7 we make use of the previous technical tools to establish the $C^{1,\gamma }$-regularity of the regular set (Theorem 3).

Finally, in Appendix A we provide an example of almost minimizers.

1.5 Notation

We introduce here some notations that are used frequently in this paper.

$B_r(x_0)$ means the open n-dimensional ball of radius r, centered at $x_0$, with boundary $\partial B_r(x_0)$.

$B_r:=B_r(0)$, $\partial B_r:=\partial B_r(0)$.

For $\textbf{u}=(u_1,\cdots ,u_m)$, $m\ge 1$, we denote

$$\begin{aligned} \textbf{u}^+&=(u_1^+,\cdots ,u_m^+),\quad \textbf{u}^-=(u_1^-,\cdots ,u_m^-),\quad \text {where }u_i^\pm =\max \{0,\pm u_i\},\\ |\textbf{u}|&=\left( \sum _{i=1}^m|u_i|^2\right) ^{1/2}. \end{aligned}$$

For a domain D, we indicate the integral mean value of $\textbf{u}$ by

In particular, when $D=B_r(x_0)$, we simply write

$$\begin{aligned} \langle {\textbf{u}}\rangle _{x_0,r}{:}{=}\langle {\textbf{u}}\rangle _{B_r(x_0)}. \end{aligned}$$

$\nabla \textbf{u}$ is an $m\times n$-matrix with its ij-th entries $\partial _{x_j}u_i$, $1\le i\le m$, $1\le j\le n$, and its norm $|\nabla \textbf{u}|=\left( \sum _{j=1}^n\sum _{i=1}^m(\partial _{x_j}u_i)^2\right) ^{1/2}$.

For a given set, $\nu$ denotes the unit outward normal to the boundary.

$\partial _\theta \textbf{u}:=\nabla \textbf{u}-(\nabla \textbf{u}\cdot \nu )\nu$ is the surface derivative of $\textbf{u}$.

$\Gamma (\textbf{u}):=\partial \{|\textbf{u}|>0\}$ is the free boundary of $\textbf{u}$.

$\Gamma ^\kappa (\textbf{u}):=\{x_0\in \Gamma (\textbf{u})\,:\, \textbf{u}(x)=O(|x-x_0|^{\xi }) \ \hbox {for some } \lfloor \kappa \rfloor<\xi <\kappa \}.$

$\lfloor s\rfloor$ is the greatest integer below $s\in {{\mathbb {R}}}$, i.e., $s-1\le \lfloor s\rfloor < s$.

For $\textbf{u}\in W^{1,2}(B_r;{{\mathbb {R}}}^m)$ and $0<q<1$, we set

$$\begin{aligned} E(\textbf{u},r):=\int _{B_r}\left( |\nabla \textbf{u}|^2+|\textbf{u}|^{q+1}\right) . \end{aligned}$$

For $\alpha$–Hölder continuous functions $\lambda _\pm :D\rightarrow {{\mathbb {R}}}^n$ satisfying $\lambda _0\le \lambda _\pm (x)\le \lambda _1$ (as in Sect. 1.1), we denote

$$\begin{aligned} f(x_0,\textbf{u})&:=\lambda _+(x_0)|\textbf{u}^+|^{q-1}\textbf{u}^+-\lambda _-(x_0)|\textbf{u}^-|^{q-1}\textbf{u}^-,\\ F(x,\textbf{u})&:=\frac{1}{1+q}\left( \lambda _+|\textbf{u}^+|^{q+1}+\lambda _-|\textbf{u}^-|^{q+1}\right) ,\\ F(x_0,\textbf{u})&:=\frac{1}{1+q}\left( \lambda _+(x_0)|\textbf{u}^+|^{q+1}+\lambda _-(x_0)|\textbf{u}^-|^{q+1}\right) ,\\ F(x_0,\textbf{u},\textbf{h})&:=\frac{1}{1+q}\left( \lambda _+(x_0)\left| |\textbf{u}^+|^{q+1}-|\textbf{h}^+|^{q+1}\right| +\lambda _-(x_0)\left| |\textbf{u}^-|^{q+1}-|\textbf{h}^-|^{q+1}\right| \right) . \end{aligned}$$

We fix constants (for $x_0\in B_{1}$)

$$\begin{aligned} \kappa :=\frac{2}{1-q},\qquad \beta _{x_0}=\lambda _+(x_0)^{\kappa /2}(\kappa (\kappa -1))^{-\kappa /2}. \end{aligned}$$

Throughout this paper, a universal constant may depend only on n, $\alpha$, M, $\kappa$ and $E(\textbf{u},1)$.

Below we consider only norms of vectorial functions to ${{\mathbb {R}}}^m$, but not those of scalar functions. Thus, for notational simplicity we drop ${{\mathbb {R}}}^m$ for spaces of vectorial functions, e.g., $C^1({{\mathbb {R}}}^n)=C^1({{\mathbb {R}}}^n;{{\mathbb {R}}}^m)$, $W^{1,2}(B_1)=W^{1,2}(B_1;{{\mathbb {R}}}^m)$.

2 Regularity of almost minimizers

The main result of this section is the $C^{1,\alpha /2}$ estimates of almost minimizers (Theorem 1). The proof is based on the Morrey and Campanato space embeddings, similar to the case of almost minimizers with $q=0$ and $\lambda _\pm =1$, treated by the authors in [2]. We first prove the following concentric ball estimates.

Proposition 1

Let $\textbf{u}$ be an almost minimizer in $B_1$. Then, there are $r_0=r_0(\alpha ,M)\in (0,1)$ and $C_0=C_0(n,M)>1$ such that

$$\begin{aligned} \int _{B_\rho (x_0)}\left( |\nabla \textbf{u}|^2+F(x_0,\textbf{u})\right)&\le C_0\left[ \left( \frac{\rho }{r}\right) ^n+r^\alpha \right] \int _{B_r(x_0)}\left( |\nabla \textbf{u}|^2+F(x_0,\textbf{u})\right) \nonumber \\&\quad +C_0r^{n+\frac{2}{1-q}(q+1-\alpha q)} \end{aligned}$$

(2.1)

for any $B_{r_0}(x_0)\Subset B_1$ and $0<\rho<r<r_0.$

Proof

Without loss of generality, we may assume $x_0=0$. Let $\textbf{h}$ be a harmonic replacement of $\textbf{u}$ in $B_r$, i.e., $\textbf{h}$ is the vectorial harmonic function with $\textbf{h}=\textbf{u}$ on $\partial B_r$. Since $|\textbf{h}^\pm |^{q+1}$ and $|\nabla \textbf{h}|^2$ are subharmonic in $B_r$, we have the following sub-mean value properties:

$$\begin{aligned} \int _{B_\rho }F(0,\textbf{h})\le \left( \frac{\rho }{r}\right) ^n\int _{B_r}F(0,\textbf{h}),\quad \int _{B_\rho }|\nabla \textbf{h}|^2\le \left( \frac{\rho }{r}\right) ^n\int _{B_r}|\nabla \textbf{h}|^2,\quad 0<\rho <r. \end{aligned}$$

(2.2)

Moreover, notice that since $\textbf{h}$ is harmonic, $\int _{B_r}\nabla \textbf{h}\cdot \nabla (\textbf{u}-\textbf{h})=0.$ Combining this with the almost minimizing property of $\textbf{u}$, we obtain that for $0<r<r_0(\alpha ,M)$,

$$\begin{aligned} \begin{aligned}&\int _{B_r}|\nabla (\textbf{u}-\textbf{h})|^2=\int _{B_r}\left( |\nabla \textbf{u}|^2-|\nabla \textbf{h}|^2\right) \\&\quad \le \int _{B_r}\left[ Mr^\alpha |\nabla \textbf{h}|^2+2(1+Mr^\alpha )F(0,\textbf{h})-2F(0,\textbf{u})\right] \\&\quad =\int _{B_r}\left[ Mr^\alpha |\nabla \textbf{h}|^2+2(1+2Mr^\alpha )(F(0,\textbf{h})-F(0,\textbf{u}))+2Mr^\alpha (2F(0,\textbf{u})-F(0,\textbf{h}))\right] \\&\quad \le \int _{B_r}\left[ Mr^\alpha |\nabla \textbf{u}|^2+3F(0,\textbf{u},\textbf{h})+2Mr^\alpha (2F(0,\textbf{u})-F(0,\textbf{h}))\right] , \end{aligned} \end{aligned}$$

(2.3)

where in the last line we have used that $F(0,\textbf{h})-F(0,\textbf{u})\le F(0,\textbf{u},\textbf{h})$.

We also note that by Poincaré inequality there is $C_1=C_1(n)>0$ such that

$$\begin{aligned} r^{-2}\int _{B_r}|\textbf{u}-\textbf{h}|^2\le C_1\int _{B_r}|\nabla (\textbf{u}-\textbf{h})|^2. \end{aligned}$$

Then, for $\varepsilon _1=\frac{1}{16C_1M}$,

$$\begin{aligned}&\int _{B_r}\frac{1}{1+q}\left| |\textbf{u}^+|^{q+1}-|\textbf{h}^+|^{q+1}\right| \\&\quad \le \int _{B_r}\left( |\textbf{u}^+|^q+|\textbf{h}^+|^q\right) |\textbf{u}^+-\textbf{h}^+|\\&\quad \le \int _{B_r}1/4\left( r^{\frac{\alpha q}{q+1}}|\textbf{u}^+|^q\right) ^{\frac{q+1}{q}}+1/4\left( r^{\frac{\alpha q}{q+1}}|\textbf{h}^+|^q\right) ^{\frac{q+1}{q}}+C\left( r^{-\frac{\alpha q}{q+1}}|\textbf{u}^+-\textbf{h}^+|\right) ^{q+1}\\&\quad =\int _{B_r}r^\alpha /4(|\textbf{u}^+|^{q+1}+|\textbf{h}^+|^{q+1})+Cr^{-\alpha q}|\textbf{u}-\textbf{h}|^{q+1}\\&\quad \le \int _{B_r}\left[ r^\alpha /4(|\textbf{u}^+|^{q+1}+|\textbf{h}^+|^{q+1})+\varepsilon _1\left( r^{-(q+1)}|\textbf{u}-\textbf{h}|^{q+1}\right) ^{\frac{2}{q+1}}+Cr^{(q+1-\alpha q)\frac{2}{1-q}}\right] \\&\quad = \int _{B_r}\left[ r^\alpha /4(|\textbf{u}^+|^{q+1}+|\textbf{h}^+|^{q+1})+\varepsilon _1 r^{-2}|\textbf{u}-\textbf{h}|^2\right] +Cr^{n+\frac{2}{1-q}(q+1-\alpha q)}\\&\quad \le \int _{B_r}\left[ r^\alpha /4(|\textbf{u}^+|^{q+1}+|\textbf{h}^+|^{q+1})+\varepsilon _1 C_1|\nabla (\textbf{u}-\textbf{h})|^2\right] +Cr^{n+\frac{2}{1-q}(q+1-\alpha q)}, \end{aligned}$$

where in the second inequality we applied Young’s inequality.

Similarly, we can get

$$\begin{aligned}&\int _{B_r}\frac{1}{1+q}||\textbf{u}^-|^{q+1}-|\textbf{h}^-|^{q+1}|\\&\quad \le \int _{B_r}\left[ r^\alpha /4(|\textbf{u}^-|^{q+1}+|\textbf{h}^-|^{q+1})+\varepsilon _1 C_1|\nabla (\textbf{u}-\textbf{h})|^2\right] +Cr^{n+\frac{2}{1-q}(q+1-\alpha q)}, \end{aligned}$$

and it follows that

$$\begin{aligned} \begin{aligned}&\int _{B_r}F(0,\textbf{u},\textbf{h})\\&\quad =\int _{B_r}\left( \frac{\lambda _+(0)}{1+q}\left| |\textbf{u}^+|^{q+1}-|\textbf{h}^+|^{q+1}\right| +\frac{\lambda _-(0)}{1+q}\left| |\textbf{u}^-|^{q+1}-|\textbf{h}^-|^{q+1}\right| \right) \\&\quad \le \int _{B_r}\big [\lambda _+(0)r^\alpha /4\left( |\textbf{u}^+|^{q+1}+|\textbf{h}^+|^{q+1}\right) +\lambda _-(0)r^\alpha /4\left( |\textbf{u}^-|^{q+1}+|\textbf{h}^-|^{q+1}\right) \\&\qquad +2\varepsilon _1 C_1M|\nabla (\textbf{u}-\textbf{h})|^2\big ]+Cr^{n+\frac{2}{1-q}(q+1-\alpha q)}\\&\quad =\int _{B_r}\left[ (1+q)r^\alpha /4(F(0,\textbf{u})+F(0,\textbf{h}))+2\varepsilon _1 C_1M|\nabla (\textbf{u}-\textbf{h})|^2\right] +Cr^{n+\frac{2}{1-q}(q+1-\alpha q)}. \end{aligned} \end{aligned}$$

(2.4)

From (2.3) and (2.4),

$$\begin{aligned}&\int _{B_r}\left( |\nabla (\textbf{u}-\textbf{h})|^2+4F(0,\textbf{u},\textbf{h})\right) \\&\quad \le \int _{B_r}\big [Mr^\alpha |\nabla \textbf{u}|^2+3F(0,\textbf{u},\textbf{h})+Cr^\alpha F(0,\textbf{u})+\left( 1+q-2M\right) r^\alpha F(0,\textbf{h})\\&\qquad +8\varepsilon _1 C_1M|\nabla (\textbf{u}-\textbf{h})|^2\big ] +Cr^{n+\frac{2}{1-q}(q+1-\alpha q)}\\&\quad \le \int _{B_r}\left[ Cr^\alpha \left( |\nabla \textbf{u}|^2+F(0,\textbf{u})\right) +3F(0,\textbf{u},\textbf{h})+\frac{1}{2}|\nabla (\textbf{u}-\textbf{h})|^2\right] \\&\qquad +Cr^{n+\frac{2}{1-q}(q+1-\alpha q)}, \end{aligned}$$

which gives

$$\begin{aligned} \int _{B_r}\left( \frac{1}{2}|\nabla (\textbf{u}-\textbf{h})|^2+F(0,\textbf{u},\textbf{h})\right)&\le Cr^\alpha \int _{B_r}\left( |\nabla \textbf{u}|^2+F(0,\textbf{u})\right) \\ {}&+Cr^{n+\frac{2}{1-q}(q+1-\alpha q)}. \end{aligned}$$

Thus

$$\begin{aligned} \begin{aligned}&\int _{B_r}\left( |\nabla (\textbf{u}-\textbf{h})|^2+F(0,\textbf{u},\textbf{h})\right) \\&\quad \le 2\int _{B_r}\left( \frac{1}{2}|\nabla (\textbf{u}-\textbf{h})|^2+F(0,\textbf{u},\textbf{h})\right) \\&\quad \le Cr^\alpha \int _{B_r}\left( |\nabla \textbf{u}|^2+F(0,\textbf{u})\right) +Cr^{n+\frac{2}{1-q}(q+1-\alpha q)}. \end{aligned} \end{aligned}$$

(2.5)

Now, by combining (2.2) and (2.5), we obtain that for $0<\rho<r<r_0$,

$$\begin{aligned}&\int _{B_\rho }\left( |\nabla \textbf{u}|^2+F(0,\textbf{u})\right) \\&\quad \le 2\int _{B_\rho }\left( |\nabla \textbf{h}|^2+F(0,\textbf{h})\right) +2\int _{B_\rho }\left( |\nabla (\textbf{u}-\textbf{h})|^2+F(0,\textbf{u},\textbf{h})\right) \\&\quad \le 2\left( \frac{\rho }{r}\right) ^n\int _{B_r}\left( |\nabla \textbf{h}|^2+F(0,\textbf{h})\right) +2\int _{B_\rho }\left( |\nabla (\textbf{u}-\textbf{h})|^2+F(0,\textbf{u},\textbf{h})\right) \\&\quad \le 4\left( \frac{\rho }{r}\right) ^n\int _{B_r}\left( |\nabla \textbf{u}|^2+F(0,\textbf{u})\right) +6\int _{B_r}\left( |\nabla (\textbf{u}-\textbf{h})|^2+F(0,\textbf{u},\textbf{h})\right) \\&\quad \le C\left[ \left( \frac{\rho }{r}\right) ^n+r^\alpha \right] \int _{B_r}\left( |\nabla \textbf{u}|^2+F(0,\textbf{u})\right) +Cr^{n+\frac{2}{1-q}(q+1-\alpha q)}. \end{aligned}$$

$\square$

From here, we deduce the almost Lipschitz regularity of almost minimizers with the help of the following lemma, whose proof can be found in [5].

Lemma 1

Let $r_0>0$ be a positive number and let $\varphi :(0,r_0)\rightarrow (0, \infty )$ be a nondecreasing function. Let a, $\beta$, and $\gamma$ be such that $a>0$, $\gamma>\beta >0$. There exist two positive numbers $\varepsilon =\varepsilon (a,\gamma ,\beta )$, $c=c(a,\gamma ,\beta )$ such that, if

$$\begin{aligned} \varphi (\rho )\le a\Bigl [\Bigl (\frac{\rho }{r}\Bigr )^{\gamma }+\varepsilon \Bigr ]\varphi (r)+b\, r^{\beta } \end{aligned}$$

for all $\rho$, r with $0<\rho \le r<r_0$, where $b\ge 0$, then one also has, still for $0<\rho<r<r_0$,

$$\begin{aligned} \varphi (\rho )\le c\Bigl [\Bigl (\frac{\rho }{r}\Bigr )^{\beta }\varphi (r)+b\rho ^{\beta }\Bigr ]. \end{aligned}$$

Theorem 4

Let $\textbf{u}$ be an almost minimizer in $B_1$. Then $\textbf{u}\in C^{0,\sigma }(B_1)$ for all $0<\sigma <1$. Moreover, for any $K\Subset B_1$,

$$\begin{aligned} \Vert \textbf{u}\Vert _{C^{0,\sigma }(K)}\le C\left( E(\textbf{u},1)^{1/2}+1\right) \end{aligned}$$

(2.6)

with $C=C(n,\alpha ,M,\sigma ,K)$.

Proof

For given $K\Subset B_1$ and $x_0\in K$, take $\delta =\delta (n,\alpha ,M,\sigma ,K)>0$ such that $\delta <\min \{r_0,{\text {dist}}(K,\partial B_1)\}$ and $\delta ^\alpha \le \varepsilon (C_0,n,n+2\sigma -2)$, where $r_0=r_0(\alpha ,M)$ and $C_0=C_0(n,M)$ are as in Proposition 1 and $\varepsilon =\varepsilon (C_0,n,n+2\sigma -2)$ is as in Lemma 1. Then, by (2.1), for $0<\rho<r<\delta$,

$$\begin{aligned} \int _{B_\rho (x_0)}\left( |\nabla \textbf{u}|^2+F(x_0,\textbf{u})\right) \le C_0\left[ \left( \frac{\rho }{r}\right) ^n+\varepsilon \right] \int _{B_r(x_0)}\left( |\nabla \textbf{u}|^2+F(x_0,\textbf{u})\right) +C_0r^{n+2\sigma -2}. \end{aligned}$$

By applying Lemma 1, we obtain

$$\begin{aligned} \int _{B_\rho (x_0)}\left( |\nabla \textbf{u}|^2+F(x_0,\textbf{u})\right) \le C\left[ \left( \frac{\rho }{r}\right) ^{n+2\sigma -2}\int _{B_r(x_0)}\left( |\nabla \textbf{u}|^2+F(x_0,\textbf{u})\right) +\rho ^{n+2\sigma -2}\right] . \end{aligned}$$

Taking $r\nearrow \delta$, we get

$$\begin{aligned} \int _{B_\rho (x_0)}\left( |\nabla \textbf{u}|^2+F(x_0,\textbf{u})\right) \le C(n,\alpha ,M,\sigma ,K)\left( E(\textbf{u},1)+1\right) \rho ^{n+2\sigma -2} \end{aligned}$$

(2.7)

for $0<\rho <\delta$. In particular, we have

$$\begin{aligned} \int _{B_\rho (x_0)}|\nabla \textbf{u}|^2\le C(n,\alpha ,M,\sigma ,K)\left( E(\textbf{u},1)+1\right) \rho ^{n+2\sigma -2}, \end{aligned}$$

and by Morrey space embedding we conclude $\textbf{u}\in C^{0,\sigma }(K)$ with

$$\begin{aligned} \Vert \textbf{u}\Vert _{C^{0,\sigma }(K)}\le C(n,\alpha ,M,\sigma ,K)\left( E(\textbf{u},1)^{1/2}+1\right) . \end{aligned}$$

$\square$

We now prove $C^{1,\alpha /2}$-regularity of almost minimizers by using their almost Lipschitz estimates above.

Proof of Theorem 1

For $K\Subset B_1$, fix a small $r_0=r_0(n,\alpha ,M,K)>0$ to be chosen later. Particularly, we ask $r_0<{\text {dist}}(K,\partial B_1)$. For $x_0\in K$ and $0<r<r_0$, let $\textbf{h}\in W^{1,2}(B_r(x_0))$ be a harmonic replacement of $\textbf{u}$ in $B_r(x_0)$. Then, by (2.5) and (2.7) with $\sigma =1-\alpha /4\in (0,1)$,

$$\begin{aligned} \begin{aligned} \int _{B_r(x_0)}|\nabla (\textbf{u}-\textbf{h})|^2&\le Cr^\alpha \int _{B_r(x_0)}\left( |\nabla \textbf{u}|^2+F(x_0,\textbf{u})\right) +Cr^{n+\frac{2}{1-q}(q+1-\alpha q)}\\&\le C\left( E(\textbf{u},1)+1\right) r^{n+\alpha /2}+Cr^{n+2}\\&\le C(n,\alpha ,M,K)\left( E(\textbf{u},1)+1\right) r^{n+\alpha /2} \end{aligned} \end{aligned}$$

(2.8)

for $0<r<r_0$. Note that since $\textbf{h}$ is harmonic in $B_r(x_0)$, for $0<\rho <r$

$$\begin{aligned} \int _{B_\rho (x_0)}|\nabla \textbf{h}-\langle {\nabla \textbf{h}}\rangle _{x_0,\rho }|^2\le \left( \frac{\rho }{r}\right) ^{n+2}\int _{B_r(x_0)}|\nabla \textbf{h}-\langle {\textbf{h}}\rangle _{x_0,r}|^2. \end{aligned}$$

Moreover, by Jensen’s inequality,

$$\begin{aligned}&\int _{B_\rho (x_0)}|\nabla \textbf{u}-\langle {\nabla \textbf{u}}\rangle _{x_0,\rho }|^2\\&\quad \le 3\int _{B_\rho (x_0)}|\nabla \textbf{h}-\langle {\nabla \textbf{h}}\rangle _{x_0,\rho }|^2+|\nabla (\textbf{u}-\textbf{h})|^2+|\langle {\nabla (\textbf{u}-\textbf{h})}\rangle _{x_0,\rho }|^2\\&\quad \le 3\int _{B_\rho (x_0)}|\nabla \textbf{h}-\langle {\nabla \textbf{h}}\rangle _{x_0,\rho }|^2+6\int _{B_\rho (x_0)}|\nabla (\textbf{u}-\textbf{h})|^2, \end{aligned}$$

and similarly,

$$\begin{aligned} \int _{B_r(x_0)}|\nabla \textbf{h}-\langle {\nabla \textbf{h}}\rangle _{x_0,r}|^2\le 3\int _{B_r(x_0)}|\nabla \textbf{u}-\langle {\nabla \textbf{u}}\rangle _{x_0,r}|^2+6\int _{B_r(x_0)}|\nabla (\textbf{u}-\textbf{h})|^2. \end{aligned}$$

Now, we use the inequalities above to obtain

$$\begin{aligned}&\int _{B_\rho (x_0)}|\nabla \textbf{u}-\langle {\nabla \textbf{u}}\rangle _{x_0,\rho }|^2\\&\quad \le 3\int _{B_\rho (x_0)}|\nabla \textbf{h}-\langle {\nabla \textbf{h}}\rangle _{x_0,\rho }|^2+6\int _{B_\rho (x_0)}|\nabla (\textbf{u}-\textbf{h})|^2\\&\quad \le 3\left( \frac{\rho }{r}\right) ^{n+2}\int _{B_r(x_0)}|\nabla \textbf{h}-\langle {\nabla \textbf{h}}\rangle _{x_0,r}|^2+6\int _{B_\rho (x_0)}|\nabla (\textbf{u}-\textbf{h})|^2\\&\quad \le 9\left( \frac{\rho }{r}\right) ^{n+2}\int _{B_r(x_0)}|\nabla \textbf{u}-\langle {\nabla \textbf{u}}\rangle _{x_0,r}|^2+24\int _{B_r(x_0)}|\nabla (\textbf{u}-\textbf{h})|^2\\&\quad \le 9\left( \frac{\rho }{r}\right) ^{n+2}\int _{B_r(x_0)}|\nabla \textbf{u}-\langle {\nabla \textbf{u}}\rangle _{x_0,r}|^2+C\left( E(\textbf{u},1)+1\right) r^{n+\alpha /2}. \end{aligned}$$

Next, we apply Lemma 1 to get

$$\begin{aligned} \int _{B_\rho (x_0)}|\nabla \textbf{u}-\langle {\nabla \textbf{u}}\rangle _{x_0,\rho }|^2&\le C\left( \frac{\rho }{r}\right) ^{n+\alpha /2}\int _{B_r(x_0)}|\nabla \textbf{u}-\langle {\nabla \textbf{u}}\rangle _{x_0,r}|^2\\&\quad +C\left( E(\textbf{u},1)+1\right) \rho ^{n+\alpha /2} \end{aligned}$$

for $0<\rho<r<r_0$. Taking $r\nearrow r_0$, we have

$$\begin{aligned} \int _{B_\rho (x_0)}|\nabla \textbf{u}-\langle {\nabla \textbf{u}}\rangle _{x_0,\rho }|^2\le C\left( E(\textbf{u},1)+1\right) \rho ^{n+\alpha /2}. \end{aligned}$$

By Campanato space embedding, we obtain $\nabla \textbf{u}\in C^{0,\alpha /4}(K)$ with

$$\begin{aligned} \Vert \nabla \textbf{u}\Vert _{C^{0,\alpha /4}(K)}\le C(E(\textbf{u},1)^{1/2}+1). \end{aligned}$$

In particular, we have

$$\begin{aligned} \Vert \nabla \textbf{u}\Vert _{L^\infty (K)}\le C(E(\textbf{u},1)^{1/2}+1) \end{aligned}$$

for any $K\Subset B_1$. With this estimate and (2.6), we can improve (2.8):

$$\begin{aligned} \int _{B_r(x_0)}|\nabla (\textbf{u}-\textbf{h})|^2&\le Cr^\alpha \int _{B_r(x_0)}\left( |\nabla \textbf{u}|^2+F(x_0,\textbf{u})\right) +Cr^{n+2}\\&\le C\left( E(\textbf{u},1)+1\right) r^{n+\alpha }+Cr^{n+2}\\&\le C\left( E(\textbf{u},1)+1\right) r^{n+\alpha }, \end{aligned}$$

and by repeating the process above we conclude that $\nabla \textbf{u}\in C^{1,\alpha /2}(K)$ with

$$\begin{aligned} \Vert \nabla \textbf{u}\Vert _{C^{0,\alpha /2}(K)}\le C(n,\alpha ,M,K)(E(\textbf{u},1)^{1/2}+1). \end{aligned}$$

$\square$

3 Weiss-type monotonicity formula

In the rest of the paper we study the free boundary of almost minimizers. This section is devoted to proving Weiss-type monotonicity formula, which is one of the most important tools in our study of the free boundary. This result is obtained from comparison with $\kappa$-homegeneous replacements, following the idea for the one in the case $q=0$ in [2].

Before stating and proving Weiss-type monotonicity formula, we introduce different types of Weiss energies that will be used in this paper. For $\kappa =\frac{2}{1-q}>2$ and $x_0, x_1\in B_{1/2}$, set

$$\begin{aligned} W^0(\textbf{v},x_0,x_1,t)&:=\frac{1}{t^{n+2\kappa -2}}\left[ \int _{B_t(x_0)}\left( |\nabla \textbf{v}|^2+2F(x_1,\textbf{v})\right) -\frac{\kappa }{t}\int _{\partial B_t(x_0)}|\textbf{v}|^2\right] ,\\ M_{x_0}(\textbf{v})&:=\int _{B_1}\left( |\nabla \textbf{v}|^2+2F(x_0,\textbf{v})\right) -\kappa \int _{\partial B_1}|\textbf{v}|^2. \end{aligned}$$

Here, we have the relation $M_{x_0}(\textbf{v})=W^0(\textbf{v},x_0,x_0,1)$. For solutions of the system, the (almost) monotonicity of $W^0$ was obtained in [4]. For example, if $\textbf{v}$ solves $\Delta \textbf{v}=f(x_1,\textbf{v})$ then $t\longmapsto W^0(\textbf{v},x_0,x_1,t)$ is monotone in t for any point $x_0$. This fact will be used in Sect. 7.

Concerning almost minimizers, we introduce the following multiplicative perturbation of $W^0$ (with $x_0=x_1$):

$$\begin{aligned} W(\textbf{u},x_0,t):=\frac{e^{at^\alpha }}{t^{n+2\kappa -2}}\left[ \int _{B_t(x_0)}\left( |\nabla \textbf{u}|^2+2F(x_0,\textbf{u})\right) -\frac{\kappa (1-bt^\alpha )}{t}\int _{\partial B_t(x_0)}|\textbf{u}|^2\right] , \end{aligned}$$

with

$$\begin{aligned} a=\frac{M(n+2\kappa -2)}{\alpha },\quad b=\frac{M(n+2\kappa )}{\alpha }. \end{aligned}$$

Theorem 5

(Weiss-type monotonicity formula) Let $\textbf{u}$ be an almost minimizer in $B_1$. For $x_0\in B_{1/2}$ and $0<t<t_0(n,\alpha ,\kappa ,M)$,

$$\begin{aligned} \frac{d}{dt}W(\textbf{u},x_0,t)\ge \frac{e^{at^\alpha }}{t^{n+2\kappa -2}}\int _{\partial B_t(x_0)}\left| \partial _\nu \textbf{u}-\frac{\kappa (1-bt^\alpha )}{t}\textbf{u}\right| ^2. \end{aligned}$$

In particular, $W(\textbf{u},x_0,t)$ is nondecreasing in t for $0<t<t_0$.

Proof

We follow the argument in Theorem 5.1 in [6]. Without loss of generality, we may assume $x_0=0$. Then, for $0<t<1/2$, define the $\kappa$-homogeneous replacement of $\textbf{u}$ in $B_t$

$$\begin{aligned} \textbf{w}(x):=\left( \frac{|x|}{t}\right) ^\kappa \textbf{u}\left( t\frac{x}{|x|}\right) ,\quad x\in B_t. \end{aligned}$$

Note that $\textbf{w}$ is homogeneous of degree $\kappa$ in $B_t$ and coincides with $\textbf{u}$ on $\partial B_t$, that is a valid competitor for $\textbf{u}$ in $B_t$. We compute

$$\begin{aligned} \int _{B_t}|\nabla \textbf{w}|^2&=\int _{B_t}\left( \frac{|x|}{t}\right) ^{2\kappa -2}\left| \frac{\kappa }{t}\textbf{u}\left( t\frac{x}{|x|}\right) \frac{x}{|x|}+\nabla \textbf{u}\left( t\frac{x}{|x|}\right) -\nabla \textbf{u}\left( t\frac{x}{|x|}\right) \cdot \frac{x}{|x|}\frac{x}{|x|}\right| ^2\\&=\int _0^t\int _{\partial B_r}\left( \frac{r}{t}\right) ^{2\kappa -2}\left| \frac{\kappa }{t}u\left( t\frac{x}{r}\right) \nu -\left( \nabla \textbf{u}\left( t\frac{x}{r}\right) \nu \right) \nu +\nabla \textbf{u}\left( t\frac{x}{r}\right) \right| ^2\,dS_x\,dr\\&=\int _0^t\int _{\partial B_t}\left( \frac{r}{t}\right) ^{n+2\kappa -3}\left| \frac{\kappa }{t}\textbf{u}\nu -(\partial _\nu \textbf{u})\nu +\nabla \textbf{u}\right| ^2\,dS_x\,dr\\&=\frac{t}{n+2\kappa -2}\int _{\partial B_t}\left| \nabla \textbf{u}-(\partial _\nu \textbf{u})\nu +\frac{\kappa }{t}\textbf{u}\nu \right| ^2\,dS_x\\&=\frac{t}{n+2\kappa -2}\int _{\partial B_t}\left( |\nabla \textbf{u}|^2-|\partial _\nu \textbf{u}|^2+\left( \frac{\kappa }{t}\right) ^2|\textbf{u}|^2\right) . \end{aligned}$$

Moreover, from

$$\begin{aligned} \int _{B_t}|\textbf{w}^\pm |^{q+1}&=\int _0^t\int _{\partial B_r}\left( \frac{r}{t}\right) ^{2\kappa -2}\left| \textbf{u}^\pm \left( \frac{t}{r}x\right) \right| ^{q+1}\,dS_x\,dr\\&=\int _0^t\int _{\partial B_t}\left( \frac{r}{t}\right) ^{n+2\kappa -3}|\textbf{u}^\pm |^{q+1}\,dS_x\,dr\\&=\frac{t}{n+2\kappa -2}\int _{\partial B_t}|\textbf{u}^\pm |^{q+1}, \end{aligned}$$

we also have

$$\begin{aligned} \int _{B_t}F(0,\textbf{w})=\frac{t}{n+2\kappa -2}\int _{\partial B_t}F(0,\textbf{u}). \end{aligned}$$

Combining those computations with the almost minimizing property of $\textbf{u}$, we get

$$\begin{aligned}&(1-Mt^\alpha )\int _{B_t}\left( |\nabla \textbf{u}|^2+2F(0,\textbf{u})\right) \\&\quad \le \frac{1}{1+Mt^\alpha }\int _{B_t}\left( |\nabla \textbf{u}|^2+2F(0,\textbf{u})\right) \le \int _{B_t}\left( |\nabla \textbf{w}|^2+2F(0,\textbf{w})\right) \\&\quad =\frac{t}{n+2\kappa -2}\int _{\partial B_t}\left( |\nabla \textbf{u}|^2-|\partial _\nu \textbf{u}|^2+\left( \frac{\kappa }{t}\right) ^2|\textbf{u}|^2+2F(0,\textbf{u})\right) . \end{aligned}$$

This gives

$$\begin{aligned} \begin{aligned}&\frac{d}{dt}\left( e^{at^\alpha }t^{-n-2\kappa +2}\right) \int _{B_t}\left( |\nabla \textbf{u}|^2+2F(0,\textbf{u})\right) \\&\quad =-(n+2\kappa -2)e^{at^\alpha }t^{-n-2\kappa +1}(1-Mt^\alpha )\int _{B_t}\left( |\nabla \textbf{u}|^2+2F(0,\textbf{u})\right) \\&\quad \ge -e^{at^\alpha }t^{-n-2\kappa +2}\int _{\partial B_t}\left( |\nabla \textbf{u}|^2-|\partial _\nu \textbf{u}|^2+\left( \frac{\kappa }{t}\right) ^2|\textbf{u}|^2+2F(0,\textbf{u})\right) . \end{aligned} \end{aligned}$$

(3.1)

Note that we can write

$$\begin{aligned} W(\textbf{u},0,t)=e^{at^\alpha }t^{-n-2\kappa +2}\int _{B_t}\left( |\nabla \textbf{u}|^2+2F(0,\textbf{u})\right) -\psi (t)\int _{\partial B_t}|\textbf{u}|^2, \end{aligned}$$

where

$$\begin{aligned} \psi (t)=\frac{\kappa e^{at^\alpha }(1-bt^\alpha )}{t^{n+2\kappa -1}}. \end{aligned}$$

Then, using (3.1), we obtain

$$\begin{aligned}&\frac{d}{dt}W(\textbf{u},0,t)\\&\quad =\frac{d}{dt}\left( e^{at^\alpha }t^{-n-2\kappa +2}\right) \int _{B_t}\left( |\nabla \textbf{u}|^2+2F(0,\textbf{u})\right) \\ {}&\qquad +e^{at^\alpha }t^{-n-2\kappa +2}\int _{\partial B_t}\left( |\nabla \textbf{u}|^2+2F(0,\textbf{u})\right) \\&\qquad -\psi '(t)\int _{\partial B_t}|\textbf{u}|^2-\psi (t)\int _{\partial B_t}\left( 2\textbf{u}\partial _\nu \textbf{u}+\frac{n-1}{t}|\textbf{u}|^2\right) \\&\quad \ge e^{at^\alpha }t^{-n-2\kappa +2}\int _{\partial B_t}|\partial _\nu \textbf{u}|^2-2\psi (t)\int _{\partial B_t}\textbf{u}\partial _\nu \textbf{u}\\&\qquad -\left( \kappa ^2e^{at^\alpha }t^{-n-2\kappa }+\psi '(t)+(n-1)\frac{\psi (t)}{t}\right) \int _{\partial B_t}|\textbf{u}|^2. \end{aligned}$$

To simplify the last term, we observe that $\psi (t)$ satisfies the inequality

$$\begin{aligned} -\frac{e^{at^\alpha }}{t^{n+2\kappa -2}}\left( \kappa ^2e^{at^\alpha }t^{-n-2\kappa }+\psi '(t)+(n-1)\frac{\psi (t)}{t}\right) -\psi (t)^2\ge 0 \end{aligned}$$

for $0<t<t_0(n,\alpha ,\kappa ,M)$. Indeed, by a direct computation, we can see that the inequality above is equivalent to

$$\begin{aligned} 2\alpha ^2-M(n+2\kappa )\left[ (n+2\kappa )(\kappa -\alpha )+2\alpha \right] t^\alpha \ge 0, \end{aligned}$$

which holds for $0<t<t_0(n,\alpha ,\kappa ,M)$. Therefore, we conclude that

$$\begin{aligned} \frac{d}{dt}W(\textbf{u},0,t)&\ge e^{at^\alpha }t^{-n-2\kappa +2}\int _{\partial B_t}|\partial _\nu \textbf{u}|^2-2\psi (t)\int _{\partial B_t}\textbf{u}\partial _\nu \textbf{u}\\&\quad +e^{-at^\alpha }t^{n+2\kappa -2}\psi (t)^2\int _{\partial B_t}|\textbf{u}|^2\\&=e^{at^\alpha }t^{-n-2\kappa +2}\int _{\partial B_t}\left| \partial _\nu \textbf{u}-e^{-at^\alpha }t^{n+2\kappa -2}\psi (t)\textbf{u}\right| ^2\\&=\frac{e^{at^\alpha }}{t^{n+2\kappa -2}}\int _{\partial B_t}\left| \partial _\nu \textbf{u}-\frac{\kappa (1-bt^\alpha )}{t}\textbf{u}\right| ^2. \end{aligned}$$

$\square$

4 Growth estimates

In this section we prove the optimal growth of almost minimizers at the free boundary (Theorem 2).

We will divide our proof into two cases:

$$\begin{aligned} \text {either}\, \kappa \not \in {{\mathbb {N}}}\,\text { or}\, \kappa \in {{\mathbb {N}}}. \end{aligned}$$

The proof for the first case $\kappa \not \in {{\mathbb {N}}}$ is given in Lemma 3, and the one for the second case $\kappa \in {{\mathbb {N}}}$ can be found in Lemma 5 and Remark 1.

We start the proof with an auxiliary result on a more general class of almost minimizers.

Lemma 2

For $0<a_0\le 1$, $0<b_0\le 1$ and $z_0\in B_{1/2}$, we define $G(z,\textbf{u}):=a_0F(b_0z+z_0,\textbf{u})$ and let $\textbf{u}$ be an almost minimizer in $B_1$ of functionals

$$\begin{aligned} \int _{B_r(z)}\left( |\nabla \textbf{u}|^2+2G(z,\textbf{u})\right) \,dx,\qquad B_r(z)\Subset B_1, \end{aligned}$$

(4.1)

with a gauge function $\omega (r)=Mr^\alpha .$ If $\textbf{u}(x)=O(|x-x_0|^\mu )$ for some $1\le \mu \le \kappa$, $\mu \not \in {{\mathbb {N}}}$, and $x_0\in B_{1/2}$, then

$$\begin{aligned} |\textbf{u}(x)|\le C_\mu |x-x_0|^\mu ,\qquad |\nabla \textbf{u}(x)|\le C_\mu |x-x_0|^{\mu -1},\quad |x-x_0|\le r_\mu \end{aligned}$$

(4.2)

with constants $C_\mu$ and $r_\mu$ depending only on $E(\textbf{u},1)$, n, $\alpha$, M, $\mu$. As before, the $O(\cdot )$ notation does not necessarily mean the uniform estimate.

Proof

We can write $G(z,\textbf{u})=\frac{1}{1+q}\left( {\tilde{\lambda }}_+(z)|\textbf{u}^+|^{q+1}+{\tilde{\lambda }}_-(z)|\textbf{u}^-|^{q+1}\right)$ for ${\tilde{\lambda }}_\pm (z)=a_0\lambda _\pm (b_0z+z_0)$, which means that $\textbf{u}$ is an almost minimizer of the energy functional (1.1) with variable coefficients ${\tilde{\lambda }}_\pm$. In the previous sections we have proved that almost minimizers with (1.5) satisfies the $C^{1,\alpha /2}$-estimate (1.6). $\textbf{u}$ also satisfies (1.5) but $1/{\tilde{\lambda }}_0\le M$ for the lower bound ${\tilde{\lambda }}_0$ of $\lambda _\pm$, since $a_0<1$. One can check, however, that in the proofs towards (1.6) the bound $1/\lambda _0\le M$ in (1.5) is used only to get the estimate for $\frac{\lambda _\pm (x)}{\lambda _\pm (x_0)}$ in (1.3) (when rewriting the almost minimizing property with variable coefficients (1.2) to frozen coefficients (1.4)). Due to cancellation $\frac{{\tilde{\lambda }}_\pm (x)}{{\tilde{\lambda }}_\pm (x_0)}=\frac{\lambda _\pm (b_0x+z_0)}{\lambda _\pm (b_0x_0+z_0)}$ satisfies (1.3), thus we can apply Theorem 1 to $\textbf{u}$ to obtain the uniform estimate

$$\begin{aligned} \Vert \textbf{u}\Vert _{C^{1,\alpha /2}(B_{1/2})}\le C(n,\alpha ,M)\left( E(\textbf{u},1)^{1/2}+1\right) . \end{aligned}$$

In view of this estimate, the statement of Lemma 2 holds for $\mu =1$.

Now we assume that the statement holds for $1\le \mu <\kappa$ and prove that it holds for $\mu +\delta \le \kappa$ with $\delta <\alpha '/2$, $\alpha '=\alpha '(n,q)\le \alpha$ small enough, and $\mu +\delta \not \in {{\mathbb {N}}}$. This will readily imply Lemma 2 by bootstrapping.

First, we claim that (4.2) implies that there exist constants $C_0>0$ and $r_0>0$, depending only on $E(\textbf{u},1)$, n, $\alpha$, M, $\mu$, $\delta$, such that for any $r\le r_0$

(4.3)

We will prove (4.3) later, and at this moment assume that it is true. Then, by Poincaré inequality (up to possibly modifying $\textbf{p}^r$ by a constant and choosing $C_0$ larger),

By a standard limiting argument, using that $s<\mu +\delta$, we obtain that for a limiting polynomial $\overline{\textbf{p}}$, and for all $r\le r_0$,

From these estimates, under the assumption $\textbf{u}(x)=O(|x|^{\mu +\delta })$ we deduce $\overline{\textbf{p}}\equiv 0$, and obtain that for all $r\le r_0$

(4.4)

On the other hand, using $\mu +\delta \le \kappa =\frac{2}{1-q}$, one can easily see that the rescalings $\textbf{v}(x):=\frac{\textbf{u}(rx)}{r^{\mu +\delta }}$, $0<r\le r_0$, are almost minimizers of the functional (4.1) with $G(z,\textbf{v})=r^{2-(1-q)(\mu +\delta )}F(rz,\textbf{v})$ and a gauge function $\omega _r(\rho )=M(r\rho )^\alpha$. This, together with (4.4), implies that the $C^{1,\alpha /2}$-estimates of $\textbf{v}$ are uniformly bounded, independent of r. This readily gives the desired estimates (4.2) for $\mu +\delta$

$$\begin{aligned} |\textbf{u}|\le C_{\mu +\delta }|x|^{\mu +\delta },\qquad |\nabla \textbf{u}|\le C_{\mu +\delta }|x|^{\mu +\delta -1}. \end{aligned}$$

We are now left with the proof of (4.3). To this aim, let $\textbf{h}$ be the harmonic replacement of $\textbf{u}$ in $B_r$. Note that $\textbf{h}$ minimizes the Dirichlet integral and attains its maximum on $\partial B_r$. Combining this with the almost-minimality of $\textbf{u}$ and (4.2) yields

(4.5)

Here, the last inequality holds for $\alpha ' \le \alpha$ small enough since $2(\mu -1)< (q+1)\mu$.

Now, in order to prove (4.3), as in standard Campanato Type estimates, it suffices to show that if (4.3) holds for r, then for a fixed constant $\rho$ small enough,

(4.6)

Indeed, since $\textbf{h}-\textbf{p}^r$ is harmonic, there exists a harmonic polynomial ${\hat{\textbf{p}}}^{\rho }$ of degree s such that

(4.7)

as long as $\rho$ is small enough, given that $s> \mu +\delta -1.$ To justify the first inequality, notice that if $\textbf{w}$ is harmonic in $B_1$ and $\textbf{q}$ is the tangent polynomial to $\textbf{w}$ at 0 of degree $s-1$ then

Thus, we are applying this inequality to $\textbf{w}=\partial _i (\textbf{h}-\textbf{p}^r)$ and $\textbf{q}=\partial _i {\hat{\textbf{p}}}^\rho$, $1\le i\le n$, with ${\hat{\textbf{p}}}^\rho$ the tangent polynomial to $\textbf{h}-\textbf{p}^r$ at 0 of degree s. The second inequality in (4.7) follows from the fact that $\textbf{h}$ is the harmonic replacement of $\textbf{u}$ in $B_r$.

From (4.5) for this specific $\rho$ for which (4.7) holds, we obtain that

Combining this inequality with (4.7), since $\delta <\alpha '/2$, we obtain the desired claim with $\textbf{p}^{r\rho }= \textbf{p}^r + {\hat{\textbf{p}}}^\rho$, as long as $C_0 \ge 12C^2_\mu \rho ^{-n -\alpha '-2(\mu -1)}.$ $\square$

Now we prove the optimal growth at free boundary points (Theorem 2) when $\kappa \not \in {\mathbb {N}}$.

Lemma 3

Let $\textbf{u}\in W^{1,2}(B_1)$ be an almost minimizer in $B_1$ and $\kappa \not \in {{\mathbb {N}}}$. Then, there exist $C>0$ and $r_0>0$, depending on n, $\alpha$, M, $\kappa$, $E(\textbf{u},1)$, such that

$$\begin{aligned} \sup _{B_r(x_0)}\left( \frac{|\textbf{u}|}{r^{\kappa }}+\frac{|\nabla \textbf{u}|}{r^{\kappa -1}}\right) \le C, \end{aligned}$$

for any $x_0\in \Gamma ^\kappa (\textbf{u})\cap B_{1/2}$ and $0<r<r_0.$

Proof

We first prove the weaker version of Lemma 3 by allowing the constants C and $r_0$ to depend on the points $x_0\in \Gamma ^\kappa (\textbf{u})\cap B_{1/2}$ as well. That is, for each $x_0\in \Gamma ^\kappa (\textbf{u})\cap B_{1/2}$,

$$\begin{aligned} \sup _{B_r(x_0)}\left( \frac{|\textbf{u}|}{r^{\kappa }}+\frac{|\nabla \textbf{u}|}{r^{\kappa -1}}\right) \le C_{x_0},\quad 0<r<r_{x_0}, \end{aligned}$$

(4.8)

where $C_{x_0}$ and $r_{x_0}$ depend on n, $\alpha$, M, $\kappa$, $E(\textbf{u},1)$ and $x_0$.

To show this weaker estimate (4.8), we assume to the contrary there is a point $x_0\in \Gamma ^\kappa (\textbf{u})\cap B_{1/2}$ and a sequence of positive radii $\{r_j\}^\infty _{j=1}\subset (0,1)$, $r_j\searrow 0$, such that

$$\begin{aligned} \sup _{B_{r_j}(x_0)}\left( \frac{|\textbf{u}|}{r_j^{\kappa }}+\frac{|\nabla \textbf{u}|}{r_j^{\kappa -1}}\right) =j,\quad \sup _{B_r(x_0)}\left( \frac{|\textbf{u}|}{r^{\kappa }}+\frac{|\nabla \textbf{u}|}{r^{\kappa -1}}\right) \le j\quad \text {for any } r_j\le r\le 1/4. \end{aligned}$$

Define the function

$$\begin{aligned} \tilde{\textbf{u}}_j(x):=\frac{\textbf{u}(r_jx+x_0)}{jr_j^{\kappa }},\quad x\in B_{\frac{1}{4r_j}}. \end{aligned}$$

Then

$$\begin{aligned} \sup _{B_1}\left( |\tilde{\textbf{u}}_j|+|\nabla \tilde{\textbf{u}}_j|\right) =1\quad \text {and } \sup _{B_R}\left( \frac{|\tilde{\textbf{u}}_j|}{R^{\kappa }}+\frac{|\nabla \tilde{\textbf{u}}_j|}{R^{\kappa -1}}\right) \le 1\quad \text {for any }1\le R\le \frac{1}{4r_j}. \end{aligned}$$

Now we claim that there exists a harmonic function $\tilde{\textbf{u}}_0\in C^1_{{\text {loc}}}({{\mathbb {R}}}^n)$ such that over a subsequence

$$\begin{aligned} \tilde{\textbf{u}}_j\rightarrow \tilde{\textbf{u}}_0\quad \text {in }C^1_{{\text {loc}}}({{\mathbb {R}}}^n). \end{aligned}$$

Indeed, for a fixed $R>1$ and a ball $B_\rho (z)\subset B_R$, we have

$$\begin{aligned} \int _{B_\rho (z)}\left( |\nabla \tilde{\textbf{u}}_j|^2+2F_j(z,\tilde{\textbf{u}}_j)\right) =\frac{1}{j^2r_j^{n+2\kappa -2}}\int _{B_{r_j\rho }(r_jz+x_0)}\left( |\nabla \textbf{u}|^2+2F(r_jz+x_0,\textbf{u})\right) \end{aligned}$$

for $F_j(z,\tilde{\textbf{u}}_j):=\frac{1}{j^{1-q}}F(r_jz+x_0,\tilde{\textbf{u}}_j)=\frac{1}{1+q}\left( (\lambda _j)_+(z)|(\tilde{\textbf{u}}_j)^+|^{q+1}+(\lambda _j)_-(z)|(\tilde{\textbf{u}}_j)^-|^{q+1}\right)$, where $(\lambda _j)_\pm (z)=\frac{1}{j^{1-q}}\lambda _\pm (r_jz+x_0)$. Since each $\tilde{\textbf{u}}_j$ is an almost minimizer of functional (4.1) with gauge function $\omega _j(\rho )\le M(r_j\rho )^\alpha \le M\rho ^\alpha$, we can apply Theorem 1 to $\tilde{\textbf{u}}_j$ to obtain

$$\begin{aligned} \Vert \tilde{\textbf{u}}_j\Vert _{C^{1,\alpha /2}(\overline{B_{R/2}})}\le C(n,\alpha ,M,R)\left( E(\tilde{\textbf{u}}_j,1)^{1/2}+1\right) \le C(n,\alpha ,M,R). \end{aligned}$$

This implies that up to a subsequence,

$$\begin{aligned} \tilde{\textbf{u}}_j\rightarrow \tilde{\textbf{u}}_0\quad \text {in }C^1(B_{R/2}). \end{aligned}$$

By letting $R\rightarrow \infty$ and using Cantor’s diagonal argument, we further have

$$\begin{aligned} \tilde{\textbf{u}}_j\rightarrow \tilde{\textbf{u}}_0\quad \text {in }C^1_{{\text {loc}}}({{\mathbb {R}}}^n). \end{aligned}$$

To show that $\tilde{\textbf{u}}_0$ is harmonic, we fix $R>1$ and observe that for the harmonic replacement $\textbf{h}_j$ of $\tilde{\textbf{u}}_j$ in $B_R$,

$$\begin{aligned} \int _{B_R}\left( |\nabla \tilde{\textbf{u}}_j|^2+\frac{2}{j^{1-q}}F(x_0,\tilde{\textbf{u}}_j)\right) \le \left( 1+M(r_jR)^\alpha \right) \int _{B_R}\left( |\nabla \textbf{h}_j|^2+\frac{2}{j^{1-q}}F(x_0,\textbf{h}_j) \right) . \end{aligned}$$

(4.9)

From the global estimates of harmonic function $\textbf{h}_j$

$$\begin{aligned} \Vert \textbf{h}_j\Vert _{C^{1,\alpha /2}(\overline{B_R})}\le C(n,R)\Vert \tilde{\textbf{u}}_j\Vert _{C^{1,\alpha /2}(\overline{B_R})}\le C(n,\alpha ,M,R), \end{aligned}$$

we see that over a subsequence

$$\begin{aligned} \textbf{h}_j\rightarrow \textbf{h}_0 \quad \text {in }C^1(\overline{B_R}), \end{aligned}$$

for some harmonic function $\textbf{h}_0\in C^1(\overline{B_R})$. Taking $j\rightarrow \infty$ in (4.9), we get

$$\begin{aligned} \int _{B_R}|\nabla \tilde{\textbf{u}}_0|^2\le \int _{B_R}|\nabla \textbf{h}_0|^2, \end{aligned}$$

which implies that $\tilde{\textbf{u}}_0$ is the energy minimizer of the Dirichlet integral, or the harmonic function. This finishes the proof of the claim.

Now, we observe that the harmonic function $\tilde{\textbf{u}}_0$ satisfies

$$\begin{aligned} \sup _{B_1}(|\tilde{\textbf{u}}_0|+|\nabla \tilde{\textbf{u}}_0|)=1 \quad \text {and }\sup _{B_R}\left( \frac{|\tilde{\textbf{u}}_0|}{R^{\kappa }}+\frac{|\nabla \tilde{\textbf{u}}_0|}{R^{\kappa -1}}\right) \le 1\quad \text {for any }R\ge 1. \end{aligned}$$

(4.10)

On the other hand, from $x_0\in \Gamma ^\kappa (\textbf{u})$, we have $\tilde{\textbf{u}}_j(x)=\frac{\textbf{u}(r_jx+x_0)}{jr_j^\kappa }=O(|x|^{\xi })$ for some $\lfloor \kappa \rfloor<\xi <\kappa$. Applying Lemma 2 yields $|\tilde{\textbf{u}}_j(x)|\le C_{\xi }|x|^{\xi }$, $|x|<r_{\xi }$, with $C_{\xi }$ and $r_{\xi }$ depending only on n, $\alpha$, M, $\xi$. In fact, as a result of Lemma 2, $C_\xi$, $r_\xi$ may depend on $E({\tilde{u}}_j,1)$ as well. However, from $\sup _{B_1}(|{\tilde{\textbf{u}}}_j|+|\nabla {\tilde{\textbf{u}}}_j|)=1$, we infer that $E({\tilde{\textbf{u}}}_j,1)=\int _{B_1}\left( |\nabla {\tilde{\textbf{u}}}_j|^2+|{\tilde{\textbf{u}}}_j|^{q+1}\right) \le C(n)$, which allow us to say that $C_\xi$, $r_\xi$ depend only on n, $\alpha$, M, $\xi$.

As ${\tilde{\textbf{u}}}_j\rightarrow {\tilde{\textbf{u}}}_0$ in $C^1_{{\text {loc}}}({{\mathbb {R}}}^n)$, we also have $|\tilde{\textbf{u}}_0(x)|\le C_{\xi }|x|^{\xi }$, $|x|<r_{\xi }$. This readily implies

$$\begin{aligned} |\tilde{\textbf{u}}_0(0)|=|\nabla \tilde{\textbf{u}}_0(0)|=\cdots =|D^{\lfloor \kappa \rfloor }\tilde{\textbf{u}}_0(0)|=0, \end{aligned}$$

which combined with (4.10) contradicts Liouville’s theorem, and (4.8) is proved.

The pointwise estimate (4.8) tells us $\textbf{u}(x)=O(|x-x_0|^\kappa )$ at every free boundary point $x_0\in \Gamma ^\kappa (\textbf{u})\cap B_{1/2}$. This in turn implies, using Lemma 2 again, the desired uniform estimate in Lemma 3. $\square$

In the rest of this section we establish the optimal growth of almost minimizers at free boundary points when $\kappa$ is an integer. We start with weak growth estimates.

Lemma 4

Let $\textbf{u}\in W^{1,2}(B_1)$ be an almost minimizer in $B_1$ and $\kappa \in {{\mathbb {N}}}$, $\kappa > 2$. Then for any $\kappa -1<\mu <\kappa$, there exist $C>0$ and $r_0>0$, depending on n, $\alpha$, M, $\mu$, $E(\textbf{u},1)$, such that

$$\begin{aligned} \sup _{B_r(x_0)}\left( \frac{|\textbf{u}|}{r^{\mu }}+\frac{|\nabla \textbf{u}|}{r^{\mu -1}}\right) \le C \end{aligned}$$

for any $x_0\in \Gamma ^\kappa (\textbf{u})\cap B_{1/2}$ and $0<r<r_0$.

Proof

The proof is similar to that of Lemma 3. $\square$

For $0<s<1$ small to be chosen later, we define the homogeneous rescaling of $\textbf{u}$

$$\begin{aligned} \textbf{u}_s(x):=\frac{\textbf{u}(sx)}{s^\kappa },\quad {x\in B_{1/s}}. \end{aligned}$$

Recall that $\textbf{u}_s$ is an almost minimizer with gauge function $\omega (r)\le M(sr)^\alpha$. By Lemma 4, we have for all $\kappa -1<\mu <\kappa$

with $C_\mu$ depending on $E(\textbf{u},1)$, $\mu$, n, $\alpha$, M.

Lemma 5

Let $\textbf{u}$ and $\kappa$ be as in Lemma 4, and $0\in \Gamma ^\kappa (\textbf{u})\cap B_{1/2}$. Assume that in a ball $B_r$, $r\le r_0$ universal, we have for universal constants $0<s<1$, $C_0>1$ and $\kappa -1<\mu <\kappa$

(4.11)

with $\textbf{p}^r$ a harmonic polynomial of degree $\kappa$ such that

$$\begin{aligned} \Vert \textbf{p}^r\Vert _{L^\infty (B_r)}\le C_0L_s^{\frac{2}{1+q}}r^\kappa . \end{aligned}$$

(4.12)

Then, there exists $\rho >0$ small universal such that (4.11) and (4.12) hold in $B_{\rho r}$ for a harmonic polynomial $\textbf{p}^{\rho r}$ of degree $\kappa$.

Remark 1

Lemma 5 readily implies Theorem 2 when $\kappa \in {\mathbb {N}}$. In fact, as in the standard Campanato Type estimates, the lemma ensures that (4.11)–(4.12) are true for small $r\le r_1$. Combining these two estimates yields

Scaling back to $\textbf{u}(x)=s^\kappa \textbf{u}_s(x/s)$ gives its optimal growth estimates at $0\in \Gamma ^\kappa (\textbf{u})\cap B_{1/2}$. This also holds for any $x_0\in \Gamma ^\kappa (\textbf{u})\cap B_{1/2}$ by considering $\textbf{u}(\cdot -x_0)$.

Proof

For notational simplicity, we write $\textbf{v}:=\textbf{u}_s$.

Step 1. For $0<r<1$, we denote by ${\tilde{\textbf{v}}}$ and ${\tilde{\textbf{p}}}^r$ the rescalings of $\textbf{v}$ and $\textbf{p}^r$, respectively, to the ball of radius r, that is

$$\begin{aligned} {\tilde{\textbf{v}}}(x):=\frac{\textbf{v}(rx)}{r^\kappa } = \frac{\textbf{u}(srx)}{(s r)^\kappa },\qquad {\tilde{\textbf{p}}}^r(x):=\frac{\textbf{p}^r(rx)}{r^\kappa },\quad x\in B_1. \end{aligned}$$

When not specified, $\Vert \cdot \Vert _\infty$ denotes the $L^\infty$ norm in the unit ball $B_1$. With these notations, (4.11)–(4.12) read

(4.13)

and

$$\begin{aligned} \Vert {\tilde{\textbf{p}}}^r\Vert _\infty \le C_0L_s^{\frac{2}{1+q}}. \end{aligned}$$

(4.14)

We claim that if ${\tilde{\textbf{p}}}^r$ is $\kappa$-homogeneous, then the finer bound

$$\begin{aligned} \Vert {\tilde{\textbf{p}}}^r\Vert _\infty \le \frac{C_0}{2}L_s^{\frac{2}{1+q}} \end{aligned}$$

holds for a universal constant $C_0>0$. Indeed, applying Theorem 5, Weiss-type monotonicity formula, gives

$$\begin{aligned} \int _{B_1}\left( |\nabla {\tilde{\textbf{v}}}|^2+2F(0,{\tilde{\textbf{v}}})\right) -\kappa (1-b(rs)^\alpha )\int _{\partial B_1}|{\tilde{\textbf{v}}}|^2\le e^{-a(rs)^\alpha }W(\textbf{u},0,t_0) \end{aligned}$$

and since $b>0$,

$$\begin{aligned} \frac{2\lambda _0}{1+q}\int _{B_1}|{\tilde{\textbf{v}}}|^{q+1}\le C-\left( \int _{B_1}|\nabla {\tilde{\textbf{v}}}|^2-\kappa \int _{\partial B_1}|{\tilde{\textbf{v}}}|^2\right) . \end{aligned}$$

Using that ${\tilde{\textbf{p}}}^r$ is a $\kappa$-homogeneous harmonic polynomial satisfying (4.13), we get

$$\begin{aligned} \frac{2\lambda _0}{1+q}\int _{B_1}|{\tilde{\textbf{v}}}|^{q+1}&\le C-\left( \int _{B_1}|\nabla ({\tilde{\textbf{v}}}-{\tilde{\textbf{p}}}^r)|^2-\kappa \int _{\partial B_1}|{\tilde{\textbf{v}}}-{\tilde{\textbf{p}}}^r|^2\right) \\&\le C(1+L_s^2). \end{aligned}$$

Thus

$$\begin{aligned} \int _{B_1}|{\tilde{\textbf{p}}}^r|^{q+1}&\le C\int _{B_1}\left( |{\tilde{\textbf{v}}}|^{q+1}+|{\tilde{\textbf{v}}}-{\tilde{\textbf{p}}}^r|^{q+1}\right) \\&\le C\left( 1+L_s^2+\Vert {\tilde{\textbf{v}}}-{\tilde{\textbf{p}}}^r\Vert _{L^2(B_1)}^{q+1}\right) \le C(1+L_s^2+L_s^{q+1}). \end{aligned}$$

In conclusion (for a universal constant $C>0$) we have $\Vert {\tilde{\textbf{p}}}^r\Vert _\infty ^{q+1}\le CL_s^2$ from which we deduce that

$$\begin{aligned} \Vert {\tilde{\textbf{p}}}^r\Vert _\infty \le \frac{C_0}{2}L_s^{\frac{2}{1+q}}. \end{aligned}$$

Step 2. We claim that for some $t_0>0$ small universal

$$\begin{aligned} |\nabla {\tilde{\textbf{v}}}(x)|\le C_\mu L_s^{\frac{2}{1+q}}|x|^{\mu -1},\,\, |{\tilde{\textbf{v}}}(x)|\le C_\mu L_s^{\frac{2}{1+q}}|x|^\mu ,\quad x\in B_{t_0}. \end{aligned}$$

(4.15)

Indeed, (4.13)–(4.14) give (C universal possibly changing from equation to equation)

Similarly,

and Höder’s inequality gives

We conclude that the following energy estimate $E( {\tilde{\textbf{w}}},1)\le C,$ where ${\tilde{\textbf{w}}}:=L_s^{-\frac{2}{1+q}}{\tilde{\textbf{v}}}$ is an almost minimizer with the same gauge function as ${\tilde{\textbf{v}}}$ for the energy

$$\begin{aligned} \int _{B_t(x_0)}\left( |\nabla {\tilde{\textbf{w}}}|^2+2L_s^{\frac{2(q-1)}{q+1}}F(x_0,{\tilde{\textbf{w}}})\right) ,\quad 0<t<1. \end{aligned}$$

As before, $L_s^{\frac{2(q-1)}{q+1}}\le 1$ allows us to repeat the arguments towards the $C^{1,\alpha /2}$-estimtate of almost minimizers as well as towards Lemma 4. Since $\textbf{u}=o(|x|^{\kappa -1})$ implies ${\tilde{\textbf{w}}}=o(|x|^{\kappa -1})$, we can apply Lemma 4 to have

$$\begin{aligned} |{\tilde{\textbf{w}}}(x)|\le C_\mu |x|^\mu ,\,\, |\nabla {\tilde{\textbf{w}}}(x)|\le C_\mu |x|^{\mu -1},\quad x\in B_{t_0}. \end{aligned}$$

This readily implies (4.15).

Step 3. Let ${\tilde{\textbf{h}}}$ be the harmonic replacement of ${\tilde{\textbf{v}}}$ in $B_{t_0}$. Then, we claim that

(4.16)

Let us first recall that $\textbf{v}(x)=\frac{\textbf{u}(sx)}{s^\kappa }$ and that ${\tilde{\textbf{v}}}(x)=\frac{\textbf{v}(rx)}{r^\kappa }=\frac{\textbf{u}(rsx)}{(rs)^\kappa }$ is an almost minimizer with gauge function $\omega (\rho )\le M(rs\rho )^\alpha$. Thus,

To estimate I, we use that ${\tilde{\textbf{h}}}$ is the harmonic replacement of ${\tilde{\textbf{v}}}$, together with (4.15), to get

In addition, it follows from the subharmonicity of $|{\tilde{\textbf{h}}}|^2$ and (4.15) that

$$\begin{aligned} \Vert {\tilde{\textbf{h}}}\Vert _{L^\infty (B_{t_0})}=\Vert {\tilde{\textbf{h}}}\Vert _{L^\infty (\partial B_{t_0})}=\Vert {\tilde{\textbf{v}}}\Vert _{L^\infty (\partial B_{t_0})}\le CL_s^{\frac{2}{1+q}}. \end{aligned}$$

This gives

Therefore,

$$\begin{aligned} I\le Cs^\alpha L_s^{\frac{4}{1+q}}=CL_s^{\frac{\alpha }{\mu -\kappa }+\frac{4}{1+q}}\le CL_s^{1+\frac{2q}{1+q}}, \end{aligned}$$

where the last inequality holds if $\mu$ is chosen universal close enough to $\kappa$ (specifically, $\mu \ge \kappa -\frac{\alpha (1+q)}{3(1-q)})$.

Next, we estimate II.

To bound the last term, we observe

$$\begin{aligned} \int _{B_{t_0}}\nabla ({\tilde{\textbf{h}}}-{\tilde{\textbf{p}}}^r)\cdot \nabla ({\tilde{\textbf{h}}}-{\tilde{\textbf{v}}})&=\int _{\partial B_{t_0}}\partial _\nu ({\tilde{\textbf{h}}}-{\tilde{\textbf{p}}}^r)({\tilde{\textbf{h}}}-{\tilde{\textbf{v}}})-\int _{B_{t_0}}\Delta ({\tilde{\textbf{h}}}-{\tilde{\textbf{p}}}^r)({\tilde{\textbf{h}}}-{\tilde{\textbf{v}}})\\&=0, \end{aligned}$$

and use it to obtain

$$\begin{aligned} \int _{B_{t_0}}|\nabla ({\tilde{\textbf{v}}}-{\tilde{\textbf{p}}}^r)|^2-\int _{B_{t_0}}|\nabla ({\tilde{\textbf{v}}}-{\tilde{\textbf{h}}})|^2&=\int _{B_{t_0}}|\nabla {\tilde{\textbf{p}}}^r|^2-|\nabla {\tilde{\textbf{h}}}|^2-2\nabla {\tilde{\textbf{v}}}\cdot \nabla ({\tilde{\textbf{p}}}^r-{\tilde{\textbf{h}}})\\&=\int _{B_{t_0}}\nabla ({\tilde{\textbf{p}}}^r-{\tilde{\textbf{h}}})\cdot \nabla ({\tilde{\textbf{p}}}^r+{\tilde{\textbf{h}}}-2{\tilde{\textbf{v}}})\\&=\int _{B_{t_0}}\nabla ({\tilde{\textbf{p}}}^r-{\tilde{\textbf{h}}})\cdot \nabla ({\tilde{\textbf{p}}}^r-{\tilde{\textbf{h}}})\\&=\int _{B_{t_0}}|\nabla ({\tilde{\textbf{p}}}^r-{\tilde{\textbf{h}}})|^2\ge 0. \end{aligned}$$

Therefore,

where we used (4.13) in the last inequality. This completes the proof of (4.16).

Step 4. For $\rho \in (0,t_0)$ small to be chosen below, we have by (4.16)

Since ${\tilde{\textbf{h}}}-{\tilde{\textbf{p}}}^r$ is harmonic, arguing as in the proof of Lemma 2, we can find a harmonic polynomial $\textbf{q}^r$ (in $B_r$) of degree $\kappa$ such that ${\tilde{\textbf{q}}}^r (x)=\frac{\textbf{q}^r (rx)}{r^\kappa }$ satisfies

Using (4.13) and (4.16), we further have

This, combined with the equation above, gives

By possibly modifying ${\tilde{\textbf{q}}}^r$ by adding a constant, we also have by Poincaré inequality

One can see that ${\tilde{\textbf{q}}}^r$ depends on $\rho$ as well as r, but we keep denoting ${\tilde{\textbf{q}}}^r$ for the notational simplicity. We choose $\rho \in (0,t_0)$ small so that

$$\begin{aligned} C_1\rho ^2\le 1/8 \end{aligned}$$

and then choose $L_s$ large (that is s small) so that

$$\begin{aligned} C_1\rho ^{-n-2\kappa +2}L_s^{\frac{q-1}{1+q}}\le 1/8. \end{aligned}$$

This yields that

(4.17)

Notice that (4.17) holds for any $\rho \in [\rho _1,\rho _2]$, with some constants $\rho _1$, $\rho _2>0$ small universal and $L_s>0$ large universal.

In addition, we have

(4.18)

where the last line follows from (4.13) and (4.17). We remark that ${\bar{C}}$ depends on $\rho _2$, but is independent of $\rho _1$.

Step 5. In this step, we prove that the estimates (4.11)–(4.12) over $B_r$ imply the same estimates over $B_{\rho r}$. We set (by abuse of notation) $\textbf{p}^{\rho r}:=\textbf{p}^r+\textbf{q}^r$ and recall $\textbf{q}^r(x):=r^\kappa {\tilde{\textbf{q}}}^r\left( \frac{x}{r}\right)$. Following the notations above we denote its homogeneous rescaling by

$$\begin{aligned} {\tilde{\textbf{p}}}^{\rho r}(x):=\frac{\textbf{p}^{\rho r}(\rho rx)}{(\rho r)^\kappa }=\frac{({\tilde{\textbf{p}}}^r+{\tilde{\textbf{q}}}^r)(\rho x)}{\rho ^\kappa }. \end{aligned}$$

We divide the proof into the following two cases:

$$\begin{aligned} \text {either }\textbf{p}^{\rho r}\text { is homogeneous of degree }\kappa \text { or not}. \end{aligned}$$

Case 1. Suppose that $\textbf{p}^{\rho r}$ is $\kappa$-homogeneous. Then (4.11) over $B_{\rho r}$ follows from (4.17) and (4.12) over $B_{\rho r}$ with $\textbf{p}^{\rho r}$ from the monotonicity formula, see the claim in Step 1.

Case 2. Now we assume that $\textbf{p}^{\rho r}$ is not homogeneous of degree $\kappa$. Note that for each polynomial $\textbf{p}$ of degree $\kappa$, we can decompose $\textbf{p}=\textbf{p}_h+\textbf{p}_i$ with $\textbf{p}_h$, $\textbf{p}_i$ respectively the $\kappa$-homogeneous and the inhomogeneous parts of $\textbf{p}$. We will prove that (4.11)–(4.12) hold in $B_{\rho r}$ with the harmonic polynomial $\textbf{p}^{\rho r}_h$ (in the place of $\textbf{p}^{\rho r}$). In fact, it is enough to prove the statement in $B_{\rho r}$ under the assumption that $\textbf{p}^r$ is $\kappa$-homogeneous. Indeed, we note that (4.17) holds for every any $r\le 1$ and $\rho _1<\rho <\rho _2$. Thus, if $r\le r_0$ is small enough, we can find $\rho \in [\rho _1,\rho _2]$ such that $r=\rho ^m$ for some $m\in {\mathbb {N}}$. Then, we can iterate the above statement with such $\rho$, starting with $\textbf{p}^1=0$.

Now, we distinguish two subcases, for $\delta >0$ small and $L_s>0$ large universal:

$$\begin{aligned} \Vert {\tilde{\textbf{q}}}^r_i\Vert _\infty \le \delta L_s\quad \text {or}\quad \Vert {\tilde{\textbf{q}}}^r_i\Vert _\infty >\delta L_s. \end{aligned}$$

Case 2.1. We first consider the case $\Vert {\tilde{\textbf{q}}}^r_i\Vert _\infty \le \delta L_s$. To prove (4.11), we use the $\kappa$-homogeneity of $\textbf{p}^r$ to have

$$\begin{aligned} \textbf{p}^{\rho r}_i=\textbf{p}^r_i+\textbf{q}^r_i=\textbf{q}^r_i, \end{aligned}$$

which implies (in accordance with the decomposition above)

$$\begin{aligned} \textbf{p}^{\rho r}_h=\textbf{p}^{\rho r}-\textbf{p}^{\rho r}_i=\textbf{p}^{\rho r}-\textbf{q}^r_i. \end{aligned}$$

Combining this with (4.17) gives

Similarly,

This proves (4.11) in $B_{\rho r}$ with harmonic polynomial $\textbf{p}_h^{\rho r}$. (4.12) follows from the homogeneity of $\textbf{p}^{\rho r}_h$.

Case 2.2. Now we assume $\Vert {\tilde{\textbf{q}}}^r_i\Vert _\infty >\delta L_s$. We will show that it leads to a contradiction and that we always fall in the previous case.

Indeed, for ${\bar{r}}:=\rho _1 r$,

$$\begin{aligned} \Vert {\tilde{\textbf{p}}}_i^{{\bar{r}}}\Vert _\infty =\left\| \frac{{\tilde{\textbf{q}}}^r_i(\rho _1x)}{\rho _1^\kappa }\right\| _\infty =\frac{\Vert {\tilde{\textbf{q}}}^r_i\Vert _{L^\infty (B_{\rho _1})}}{\rho _1^\kappa }\ge \frac{\Vert {\tilde{\textbf{q}}}^r_i\Vert _{L^\infty (B_1)}}{\rho _1}\ge \frac{\delta }{\rho _1}L_s. \end{aligned}$$

Recall that the constant ${\bar{C}}$ in (4.18) is independent of $\rho _1$. Thus, for $\rho _1>0$ small,

$$\begin{aligned} \Vert {\tilde{\textbf{p}}}^{{\bar{r}}}_i\Vert _\infty \ge C_3L_s,\quad C_3\ge 2{\bar{C}}. \end{aligned}$$

Similarly,

$$\begin{aligned} \Vert {\tilde{\textbf{p}}}^{{\bar{r}}}_i\Vert _\infty =\frac{\Vert {\tilde{\textbf{q}}}^r_i\Vert _{L^\infty (B_{\rho _1})}}{\rho _1^\kappa }\le \frac{\Vert {\tilde{\textbf{q}}}^r\Vert _{L^\infty (B_{1})}}{\rho _1^\kappa }\le C_4(\rho _1)L_s. \end{aligned}$$

For $L_s$ large

$$\begin{aligned} \Vert {\tilde{\textbf{p}}}^{{\bar{r}}}_h\Vert _\infty =\Vert {\tilde{\textbf{p}}}^r+{\tilde{\textbf{q}}}^r_h\Vert _\infty \le C_0L_s^{\frac{2}{1+q}}+{\bar{C}}L_s\le \frac{3}{4}C_0L_s^{\frac{2}{1+q}}, \end{aligned}$$

and thus

$$\begin{aligned} \Vert {\tilde{\textbf{p}}}^{{\bar{r}}}\Vert _\infty \le \Vert {\tilde{\textbf{p}}}^{{\bar{r}}}_h\Vert _\infty +\Vert {\tilde{\textbf{p}}}^{{\bar{r}}}_i\Vert _\infty \le \frac{3}{4}C_0L_s^{\frac{2}{1+q}}+C_4L_s\le \frac{7}{8}C_0L_s^{\frac{2}{1+q}}. \end{aligned}$$

We iterate again, and conclude that

$$\begin{aligned} \frac{\rho _1^{-1}}{2}\Vert {\tilde{\textbf{p}}}^{{\bar{r}}}_i\Vert _\infty \le \Vert {\tilde{\textbf{p}}}^{\rho _1{\bar{r}}}_i\Vert _\infty \le 2\rho _1^{-\kappa }\Vert {\tilde{\textbf{p}}}^{{\bar{r}}}_i\Vert _\infty \end{aligned}$$

while

$$\begin{aligned} \Vert {\tilde{\textbf{p}}}^{\rho _1{\bar{r}}}_h-{\tilde{\textbf{p}}}^{{\bar{r}}}_h\Vert _\infty \le {\bar{C}}L_s. \end{aligned}$$

Indeed, using that $\Vert {\tilde{\textbf{p}}}^{{\bar{r}}}_i\Vert _\infty \ge C_3L_s\ge 2\Vert {\tilde{\textbf{q}}}^{{\bar{r}}}\Vert _\infty$, we get

$$\begin{aligned} \Vert {\tilde{\textbf{p}}}^{\rho _1{\bar{r}}}_i\Vert _\infty&=\frac{\Vert {\tilde{\textbf{p}}}^{{\bar{r}}}_i+{\tilde{\textbf{q}}}^{{\bar{r}}}_i\Vert _{L^\infty (B_{\rho _1})}}{\rho _1^\kappa }\ge \frac{\Vert {\tilde{\textbf{p}}}^{{\bar{r}}}_i+{\tilde{\textbf{q}}}^{{\bar{r}}}_i\Vert _{L^\infty (B_1)}}{\rho _1}\ge \frac{\Vert {\tilde{\textbf{p}}}^{{\bar{r}}}_i\Vert _\infty -\Vert {\tilde{\textbf{q}}}^{{\bar{r}}}\Vert _\infty }{\rho _1}\\&\ge \frac{\rho _1^{-1}}{2}\Vert {\tilde{\textbf{p}}}^{{\bar{r}}}_i\Vert _\infty , \end{aligned}$$

and similarly

$$\begin{aligned} \Vert {\tilde{\textbf{p}}}^{\rho _1{\bar{r}}}_i\Vert _\infty \le \frac{\Vert {\tilde{\textbf{p}}}^{{\bar{r}}}_i\Vert _\infty +\Vert {\tilde{\textbf{q}}}^{{\bar{r}}}\Vert _\infty }{\rho _1^\kappa }\le 2\rho _1^{-\kappa }\Vert {\tilde{\textbf{p}}}^{{\bar{r}}}_i\Vert _\infty . \end{aligned}$$

In addition,

$$\begin{aligned} \Vert {\tilde{\textbf{p}}}^{\rho _1{\bar{r}}}_h-{\tilde{\textbf{p}}}^{{\bar{r}}}_h\Vert _\infty =\Vert {\tilde{\textbf{q}}}^{{\bar{r}}}_h\Vert _\infty \le {\bar{C}}L_s. \end{aligned}$$

In conclusion, if $a_l:=\Vert {\tilde{\textbf{p}}}^{\rho _1^l{\bar{r}}}_i\Vert _\infty$ and $b_l:=\Vert {\tilde{\textbf{p}}}^{\rho _1^l{\bar{r}}}_h\Vert _\infty$, $l\ge 0$, we can iterate and have that ($\rho _1$ small)

$$\begin{aligned} C_3L_s&\le a_0\le C_4L_s,\quad 2a_l\le a_{l+1}\le C^*(\rho _1)a_l,\\ b_0&\le \frac{3}{4}C_0L_s^{\frac{2}{1+q}},\quad |b_{l+1}-b_l|\le {\bar{C}}L_s, \end{aligned}$$

as long as

$$\begin{aligned} a_l+b_l\le C_0L_s^{\frac{2}{1+q}}, \end{aligned}$$

which holds at $l=0$. This iteration is possible, since we can repeat Step 1–4 as long as $\Vert {\tilde{\textbf{p}}}^{\rho _1^l{\bar{r}}}\Vert _\infty \le a_l+b_l\le C_0L_s^{\frac{2}{1+q}}$.

Thus we can iterate till the first $l\ge 1$ ($l \sim \log L_s)$ such that for $c_0\le \frac{1}{2C^*}$ small universal

$$\begin{aligned}&\frac{C_0}{8}L_s^{\frac{2}{1+q}}\ge a_l\ge c_0L_s^{\frac{2}{1+q}}, \end{aligned}$$

(4.19)

$$\begin{aligned}&b_l\le \frac{7}{8}C_0L_s^{\frac{2}{1+q}}. \end{aligned}$$

(4.20)

We will now finally show that these inequalities lead to a contradiction.

For simplicity we write ${\tilde{\textbf{v}}}^l:={\tilde{\textbf{v}}}^{\rho _1^l{\bar{r}}}$ and ${\tilde{\textbf{p}}}^l:={\tilde{\textbf{p}}}^{\rho _1^l{\bar{r}}}$. From

we see that for any $\eta \le 1$,

Moreover, from (4.15) and (4.20), for $\mu > \kappa -1,$ and $|x| \le 1/2,$

$$\begin{aligned} L_s^{-\frac{2}{1+q}}|{\tilde{\textbf{v}}}^l - {\tilde{\textbf{p}}}_h^l|(x) \le C |x|^\mu + \frac{7}{8}C_0 |x|^\kappa . \end{aligned}$$

This combined with the equation above gives that

Hence, we conclude from (4.19) that

Since $\mu > \kappa -1$, for $\eta$ small and $L_s$ large we obtain a contradiction. $\square$

Before closing this section, we notice that the combination of Theorems 1 and 2 provides the following optimal regularity of an almost minimizer at the free boundary.

Corollary 1

Let $\textbf{u}$ be an almost minimizer in $B_1$. Then, for $x_0\in \Gamma ^\kappa (\textbf{u})\cap B_{1/2}$, $0<r<1/2$, $\textbf{u}_{x_0,r}\in C^{1,\alpha /2}(B_1)$ and for any $K\Subset B_1$,

$$\begin{aligned} \Vert \textbf{u}_{x_0,r}\Vert _{C^{1,\alpha /2}(K)}\le C(n,\alpha ,M, \kappa , K,E(\textbf{u},1)). \end{aligned}$$

(4.21)

5 Non-degeneracy

In this section we shall derive an important non-degeneracy property of almost minimizers, Theorem 6.

In the rest of this paper, for $x_0\in B_{1/2}$ and $0<r<1/2$ we denote the $\kappa$-homogeneous rescalings of $\textbf{u}$ by

$$\begin{aligned} \textbf{u}_{x_0,r}(x):=\frac{\textbf{u}(rx+x_0)}{r^\kappa },\quad x\in B_{1/(2r)}. \end{aligned}$$

Theorem 6

(Non-Degeneracy) Let $\textbf{u}$ be an almost minimizer in $B_1$. There exist constants $c_0=c_0(q,n,\alpha ,M, E(\textbf{u}, 1))>0$ and $r_0=r_0(q,n,\alpha ,M)>0$ such that if $x_0\in \Gamma ^\kappa (\textbf{u})\cap B_{1/2}$ and $0<r<r_0$, then

$$\begin{aligned} \sup _{B_{r}(x_0)}|\textbf{u}|\ge c_0r^\kappa . \end{aligned}$$

To establish Theorem 6, we first prove Lemma 6 below. The idea of this lemma is to replace the almost minimizer $\textbf{u}_{x_0,r}$ with the solution of the system with frozen coefficient at $x_0$ and boundary datum $\textbf{u}$ and use its non-degeneracy property.

Lemma 6

Let $\textbf{u}$ be an almost minimizer in $B_1$. Then, there exist small constants $\varepsilon _0=\varepsilon _0(q,n,M)>0$ and $r_0=r_0(q,n,\alpha ,M)>0$ such that for $0<r<r_0$ and $x_0\in B_{1/2}$, if $E(\textbf{u}_{x_0,r},1)\le \varepsilon _0$ then $E(\textbf{u}_{x_0,r/2},1)\le \varepsilon _0$.

Proof

For simplicity we may assume $x_0=0$. For $0<r<r_0$ to be specified later, let $\textbf{v}_r$ be a solution of $\Delta \textbf{v}_r=f(0,\textbf{v}_r)$ in $B_1$ with $\textbf{v}_r=\textbf{u}_r$ on $\partial B_1$. We claim that if $\varepsilon _0=\varepsilon _0(q,n,M)>0$ is small, then $\textbf{v}_r\equiv \textbf{0}$ in $B_{1/2}$. Indeed, if not, then $\sup _{B_{3/4}}|\textbf{v}_r|\ge c_0(q,n)$ by the non-degeneracy of the solution $\textbf{v}_r$. Thus $|\textbf{v}_r(z_0)|\ge c_0(q,n)$ for some $z_0\in \overline{B_{3/4}}$. Moreover, from $1/M\le \lambda _\pm \le M$ and $0<q<1$, we have $1/M\le \frac{2}{1+q}\lambda _\pm \le 2M$, thus

$$\begin{aligned} 1/M|\textbf{v}_r|^{q+1}\le 2F(0,\textbf{v}_r)\quad \text {and}\quad 2F(0,\textbf{u}_r)\le 2M|\textbf{u}_r|^{q+1}. \end{aligned}$$

Then

$$\begin{aligned} E(\textbf{v}_r,1)&\le M\int _{B_1}\left( |\nabla \textbf{v}_r|^2+2F(0,\textbf{v}_r)\right) \le M\int _{B_1}\left( |\nabla \textbf{u}_r|^2+2F(0,\textbf{u}_r)\right) \\ {}&\le 2M^2E(\textbf{u}_r,1)\le 2M^2\varepsilon _0. \end{aligned}$$

Combining this with the estimate for the solution $\textbf{v}_r$ gives

$$\begin{aligned} \sup _{B_{7/8}}|\nabla \textbf{v}_r|\le C(n,M)\left( E(\textbf{v}_r,1)^{1/2}+1\right) \le C(n,M), \end{aligned}$$

hence

$$\begin{aligned} |\textbf{v}_r|\ge \frac{c_0(q,n)}{2}\quad \text {in}\quad B_{\rho _0}(z_0) \end{aligned}$$

for some small $\rho _0=\rho _0(q,n,M)>0$. This implies that

$$\begin{aligned} c(q,n,M)\le \int _{B_{\rho _0}(z_0)}|\textbf{v}_r|^{q+1}\le E(\textbf{v}_r,1)\le 2M^2\varepsilon _0, \end{aligned}$$

which is a contradiction if $\varepsilon _0=\varepsilon _0(q,n,M)$ is small.

Now, we use $E(\textbf{v}_r,1)\le 2M^2\varepsilon _0$ together with the fact that $\textbf{u}_r$ satisfies (1.4) in $B_1$ with gauge function $\omega _r(\rho )\le M(r\rho )^\alpha$ to get

$$\begin{aligned} \int _{B_1}\left( |\nabla \textbf{u}_r|^2+2F(0,\textbf{u}_r)\right)&\le (1+ Mr^\alpha )\int _{B_1}\left( |\nabla \textbf{v}_r|^2+2F(0,\textbf{v}_r)\right) \\&\le 4M^4\varepsilon _0 r^\alpha +\int _{B_1}\left( |\nabla \textbf{v}_r|^2+2F(0,\textbf{v}_r)\right) , \end{aligned}$$

thus

$$\begin{aligned}&\int _{B_1}\left( |\nabla \textbf{u}_r|^2-|\nabla \textbf{v}_r|^2\right) \\&\quad \le 4M^4\varepsilon _0 r^\alpha +2\int _{B_1}\left( F(0,\textbf{v}_r)-F(0,\textbf{u}_r)\right) \\&\quad =4M^4\varepsilon _0r^\alpha +\frac{2\lambda _+(0)}{1+q}\int _{B_1}\left( |\textbf{v}_r^+|^{q+1}-|\textbf{u}_r^+|^{q+1}\right) +\frac{2\lambda _-(0)}{1+q}\int _{B_1}\left( |\textbf{v}_r^-|^{q+1}-|\textbf{u}_r^-|^{q+1}\right) . \end{aligned}$$

This gives

$$\begin{aligned} \begin{aligned}&\int _{B_1}|\nabla (\textbf{u}_r-\textbf{v}_r)|^2 =\int _{B_1}\left( |\nabla \textbf{u}_r|^2-|\nabla \textbf{v}_r|^2-2\nabla (\textbf{u}_r-\textbf{v}_r)\cdot \nabla \textbf{v}_r\right) \\&\quad =\int _{B_1}\left( |\nabla \textbf{u}_r|^2-|\nabla \textbf{v}_r|^2\right) +2\int _{B_1}(\textbf{u}_r-\textbf{v}_r)f(0,\textbf{v}_r)\\&\quad =\int _{B_1}\left( |\nabla \textbf{u}_r|^2-|\nabla \textbf{v}_r|^2\right) +2\lambda _+(0)\int _{B_1}|\textbf{v}_r^+|^{q-1} (\textbf{u}_r-\textbf{v}_r)\cdot \textbf{v}_r^+\\&\qquad -2\lambda _-(0)\int _{B_1}|\textbf{v}_r|^{q-1}(\textbf{u}_r-\textbf{v}_r)\cdot \textbf{v}_r^-\\&\quad \le 4M^4\varepsilon _0r^\alpha +\frac{2\lambda _+(0)}{1+q}\int _{B_1}\left( |\textbf{v}_r^+|^{q+1} -|\textbf{u}_r^+|^{q+1}+(1+q)|\textbf{v}_r^+|^{q-1}(\textbf{u}_r-\textbf{v}_r)\cdot \textbf{v}_r^+\right) \\&\qquad +\frac{2\lambda _-(0)}{1+q}\int _{B_1}\left( |\textbf{v}_r^-|^{q+1}-|\textbf{u}_r^-|^{q+1}-(1+q) |\textbf{v}_r^-|^{q-1}(\textbf{u}_r-\textbf{v}_r)\cdot \textbf{v}_r^-\right) \\&\quad =4M^4\varepsilon _0r^\alpha +\frac{2\lambda _+(0)}{1+q}\int _{B_1}\left( (1+q)|\textbf{v}_r^+|^{q-1}\textbf{u}_r\cdot \textbf{v}_r^+-q|\textbf{v}_r^+|^{q+1}-|\textbf{u}_r^+|^{q+1}\right) \\&\qquad +\frac{2\lambda _-(0)}{1+q}\int _{B_1}\left( -(1+q)|\textbf{v}_r^-|^{q-1}\textbf{u}_r\cdot \textbf{v}_r^--q|\textbf{v}_r^-|^{q+1} -|\textbf{u}_r^-|^{q+1}\right) . \end{aligned} \end{aligned}$$

(5.1)

To compute the last two terms, we use $\textbf{u}_r\cdot \textbf{v}_r^+\le \textbf{u}_r^+\cdot \textbf{v}_r^+\le |\textbf{u}_r^+||\textbf{v}_r^+|$ and Young’s inequality to get

$$\begin{aligned} |\textbf{v}_r^+|^{q-1}\textbf{u}_r\cdot \textbf{v}_r^+\le |\textbf{u}_r^+||\textbf{v}_r^+|^q\le \frac{1}{q+1}|\textbf{u}_r^+|^{q+1}+\frac{q}{1+q}|\textbf{v}_r^+|^{q+1}. \end{aligned}$$

Similarly, from $-\textbf{u}_r\cdot \textbf{v}_r^-\le \textbf{u}_r^-\cdot \textbf{v}_r^-\le |\textbf{u}_r^-||\textbf{v}_r^-|,$ we also have

$$\begin{aligned} -|\textbf{v}_r^-|^{q-1}\textbf{u}_r\cdot \textbf{v}_r^-\le |\textbf{u}_r^-||\textbf{v}_r^-|^q\le \frac{1}{q+1}|\textbf{u}_r^-|^{q+1}+\frac{q}{1+q}|\textbf{v}_r^-|^{q+1}. \end{aligned}$$

Combining those inequalities and (5.1) yields

$$\begin{aligned} \int _{B_1}|\nabla (\textbf{u}_r-\textbf{v}_r)|^2\le 4M^4\varepsilon _0r^\alpha . \end{aligned}$$

Applying Poincaré inequality and Hölder’s inequality, we obtain

$$\begin{aligned} \int _{B_1}\left( |\nabla (\textbf{u}_r-\textbf{v}_r)|^2+|\textbf{u}_r-\textbf{v}_r|^{q+1}\right) \le C(q,n,M)r^{\alpha /2}. \end{aligned}$$

Since $\textbf{v}_r\equiv {\textbf{0}}$ in $B_{1/2}$, we see that for $0<r<r_0(q,n,\alpha ,M)$,

$$\begin{aligned} E(\textbf{u}_r,1/2)=\int _{B_{1/2}}\left( |\nabla \textbf{u}_r|^2+|\textbf{u}_r|^{q+1}\right) \le C(q,n,M)r^{\alpha /2}\le \frac{\varepsilon _0}{2^{n+2\kappa -2}}. \end{aligned}$$

Therefore, we conclude that

$$\begin{aligned} E(\textbf{u}_{r/2},1)=2^{n+2\kappa -2}E(\textbf{u}_r,1/2)\le \varepsilon _0. \end{aligned}$$

$\square$

Lemma 6 immediately implies the following integral form of non-degeneracy.

Lemma 7

Let $\textbf{u}$, $\varepsilon _0$ and $r_0$ be as in the preceeding lemma. If $x_0\in \overline{\{|\textbf{u}|>0\}}\cap B_{1/2}$ and $0<r<r_0$, then

$$\begin{aligned} \int _{B_r(x_0)}\left( |\nabla \textbf{u}|^2+|\textbf{u}|^{q+1}\right) \ge \varepsilon _0 r^{n+2\kappa -2}. \end{aligned}$$

(5.2)

Proof

By the continuity of $\textbf{u}$, it is enough to prove (5.2) for $x_0\in \{|\textbf{u}|>0\}\cap B_{1/2}$. Towards a contradiction, we suppose that $\int _{B_r(x_0)}\left( |\nabla \textbf{u}|^2+|\textbf{u}|^{q+1}\right) \le \varepsilon _0r^{n+2\kappa -2}$, or equivalently $E(\textbf{u}_{x_0,r},1)\le \varepsilon _0$. Then, by the previous lemma we have $E(\textbf{u}_{x_0,r/2^j},1)\le \varepsilon _0$ for all $j\in {{\mathbb {N}}}$. From $|\textbf{u}(x_0)|>0$, we see that $|\textbf{u}|>c_1>0$ in $B_{r/2^j}(x_0)$ for large j. Therefore,

$$\begin{aligned} \varepsilon _0&\ge E(\textbf{u}_{x_0,r/2^j},1)=\frac{1}{(r/2^j)^{n+2\kappa -2}}\int _{B_{r/2^j}(x_0)}\left( |\nabla \textbf{u}|^2+|\textbf{u}|^{q+1}\right) \\&\ge \frac{1}{(r/2^j)^{n+2\kappa -2}}\int _{B_{r/2^j}(x_0)}2c_1^{q+1}= \frac{C(n)c_1^{q+1}}{(r/2^j)^{2\kappa -2}}\rightarrow \infty \quad \text {as }j\rightarrow \infty . \end{aligned}$$

This is a contradiction, as desired. $\square$

We are now ready to prove Theorem 6.

Proof of Theorem 6

Assume by contradiction that

$$\begin{aligned} {|\textbf{u}_{x_0,r}(x)| < c_0,} \quad x\in B_{1}, \end{aligned}$$

with $c_0$ small, to be made precise later. Let $\epsilon _0, r_0$ be the constants in Lemma 6 and $\omega _n=|B_1|$ be the volume of an n-dimensional ball. For $r<r_0$, by interpolation together with the estimate (4.21),

$$\begin{aligned} \Vert \nabla \textbf{u}_{x_0,r}\Vert _{L^\infty (B_{1/2})}&\le \epsilon \Vert \textbf{u}_{x_0,r}\Vert _{C^{1,\alpha /2}(B_{3/4})} + K(\epsilon ) \Vert \textbf{u}_{x_0,r}\Vert _{L^\infty (B_{3/4})}\\&\le \epsilon C(n,\alpha ,M, E(\textbf{u}, 1)) + K(\epsilon )c_0 \le \frac{\epsilon _0}{2^{n+2\kappa -1}\omega _n^{1/2}}, \end{aligned}$$

by choosing $\epsilon = \frac{\epsilon _0}{2^{n+2\kappa }C\omega _n^{1/2}}$ and $c_0 \le \epsilon _0/(2^{n+2\kappa }K(\epsilon )\omega _n^{1/2})$. Thus, if $\omega _n c_0^{q+1} < \epsilon _0/2^{n+2\kappa -1}$, then $E(\textbf{u}_{x_0,r}, \frac{1}{2}) < \frac{\epsilon _0}{2^{n+2\kappa -2}}$, which contradicts Lemma 7. $\square$

6 Homogeneous blowups and energy decay estimates

In this section we study the homogeneous rescalings and blowups. We first show that the $\kappa$-homogeneous blowups exist at free boundary points.

Lemma 8

Suppose $\textbf{u}$ is an almost minimizer in $B_1$ and $x_0\in \Gamma ^\kappa (\textbf{u})\cap B_{1/2}$. Then for $\kappa$-homogeneous rescalings $\textbf{u}_{x_0,t}(x)=\frac{\textbf{u}(x_0+tx)}{t^\kappa }$, there exists $\textbf{u}_{x_0,0}\in C^1_{{\text {loc}}}({{\mathbb {R}}}^n)$ such that over a subsequence $t=t_j\rightarrow 0+$,

$$\begin{aligned} \textbf{u}_{x_0,t_j}\rightarrow \textbf{u}_{x_0,0}\quad \text {in }C^1_{{\text {loc}}}({{\mathbb {R}}}^n). \end{aligned}$$

Moreover, $\textbf{u}_{x_0,0}$ is a nonzero $\kappa$-homogeneous global solution of $\Delta \textbf{u}_{x_0,0}=f(x_0,\textbf{u}_{x_0,0})$.

Proof

For simplicity we assume $x_0=0$ and write $\textbf{u}_t=\textbf{u}_{0,t}$ and $W(\textbf{u},r)=W(\textbf{u},0,0,r)$.

Step 1. We first prove the $C^1$-convergence. For any $R>1$, Corollary 1 ensures that there is a function $\textbf{u}_0\in C^1(B_{R/2})$ such that over a subsequence $t=t_j\rightarrow 0+$,

$$\begin{aligned} \textbf{u}_{t_j}\rightarrow \textbf{u}_0\quad \text {in }C^1(B_{R/2}). \end{aligned}$$

By letting $R\rightarrow \infty$ and using a Cantor’s diagonal argument, we obtain that for another subsequence $t=t_j\rightarrow 0+$,

$$\begin{aligned} \textbf{u}_{t_j}\rightarrow \textbf{u}_0\quad \text {in }C^1_{{\text {loc}}}({{\mathbb {R}}}^n). \end{aligned}$$

Step 2. By the non-degeneracy in Theorem 6, $\sup _{B_1}|\textbf{u}_{t_j}|\ge c_0>0$. By the $C^1$-convergence of $\textbf{u}_{t_j}$ to $\textbf{u}_0$, we have $\textbf{u}_0$ is nonzero. To show that $\textbf{u}_0$ is a global solution, for fixed $R>1$ and small $t_j$, let $\textbf{v}_{t_j}$ be the solution of $\Delta \textbf{v}_{t_j}=f(0,\textbf{v}_{t_j})$ in $B_R$ with $\textbf{v}_{t_j}=\textbf{u}_{t_j}$ on $\partial B_R$. Then, by elliptic theory,

$$\begin{aligned} \Vert \textbf{v}_{t_j}\Vert _{C^{1,\alpha /2}(\overline{B_R})}\le C(n,m,R)(\Vert \textbf{u}_{t_j}\Vert _{C^{1,\alpha /2}(\overline{B_R})}+1)\le C. \end{aligned}$$

Thus, there exists a solution $\textbf{v}_0\in C^1(\overline{B_R})$ such that

$$\begin{aligned} \textbf{v}_{t_j}\rightarrow \textbf{v}_0\quad \text {in }C^1(\overline{B_R}). \end{aligned}$$

Moreover, we use again that $\textbf{u}_{t_j}$ is an almost minimizer with the frozen coefficients in $B_{1/2t_j}$ with a gauge function $\omega _j(\rho )\le M(t_j\rho )^\alpha$ to have

$$\begin{aligned} \int _{B_R}\left( |\nabla \textbf{u}_{t_j}|^2+2F(0,\textbf{u}_{t_j})\right) \le (1+M(t_jR)^\alpha )\int _{B_R}\left( |\nabla \textbf{v}_{t_j}|^2+2F(0,\textbf{v}_{t_j})\right) . \end{aligned}$$

By taking $t_j\rightarrow 0$ and using the $C^1$-convergence of $\textbf{u}_{t_j}$ and $\textbf{v}_{t_j}$, we obtain

$$\begin{aligned} \int _{B_R}\left( |\nabla \textbf{u}_0|^2+2F(0,\textbf{u}_0)\right) \le \int _{B_R}\left( |\nabla \textbf{v}_0|^2+2F(0,\textbf{v}_0)\right) . \end{aligned}$$

Since $\textbf{v}_{t_j}=\textbf{u}_{t_j}$ on $\partial B_R$, we also have $\textbf{v}_0=\textbf{u}_0$ on $\partial B_R$. This means that $\textbf{u}_0$ is equal to the energy minimizer (or solution) $\textbf{v}_0$ in $B_R$. Since $R>1$ is arbitrary, we conclude that $\textbf{u}_0$ is a global solution.

Step 3. Now we prove that $\textbf{u}_0$ is $\kappa$-homogeneous. Fix $0<r<R<\infty$ and write for simplicity $W(\textbf{u},s)=W(\textbf{u},0,s)$. By the Weiss-type monotonicity formula, Theorem 5, we have that for small $t_j$,

$$\begin{aligned} \begin{aligned}&W(\textbf{u},Rt_j)-W(\textbf{u},rt_j)\\&\quad =\int _{rt_j}^{Rt_j}\frac{d}{d\rho }W(\textbf{u},\rho )\,d\rho \\&\quad \ge \int _{rt_j}^{Rt_j}\frac{1}{\rho ^{n+2\kappa }}\int _{\partial B_\rho }|x\cdot \nabla \textbf{u}-\kappa (1-b\rho ^\alpha )\textbf{u}|^2\,dS_xd\rho \\&\quad =\int _r^R\frac{t_j}{(t_j\sigma )^{n+2\kappa }}\int _{\partial B_{t_j\sigma }}|x\cdot \nabla \textbf{u}-\kappa (1-b(t_j\sigma )^\alpha )\textbf{u}|^2\,dS_xd\sigma \\&\quad =\int _r^R\frac{1}{t_j^{2\kappa }\sigma ^{n+2\kappa }}\int _{\partial B_\sigma }|t_jx\cdot \nabla \textbf{u}(t_jx)-\kappa (1-b(t_j\sigma )^\alpha )\textbf{u}(t_jx)|^2\,dS_xd\sigma \\&\quad =\int _r^R\frac{1}{\sigma ^{n+2\kappa }}\int _{\partial B_\sigma }|x\cdot \nabla \textbf{u}_{t_j}-\kappa (1-b(t_j\sigma )^\alpha )\textbf{u}_{t_j}|^2\,dS_xd\sigma . \end{aligned} \end{aligned}$$

(6.1)

On the other hand, by the optimal growth estimates Theorem 2,

$$\begin{aligned} |W(\textbf{u},r)|\le C\quad \text {for }0<r<r_0, \end{aligned}$$

thus $W(\textbf{u},0+)$ is finite. Using this and taking $t_j\rightarrow 0+$ in (6.1), we get

$$\begin{aligned} 0=W(\textbf{u},0+)-W(\textbf{u},0+)\ge \int _r^R\frac{1}{\sigma ^{n+2\kappa -1}}\int _{\partial B_\sigma }|x\cdot \nabla \textbf{u}_0-\kappa \textbf{u}_0|^2\,dS_xd\sigma . \end{aligned}$$

Taking $r\rightarrow 0+$ and $R\rightarrow \infty$, we conclude that $x\cdot \nabla \textbf{u}_0-\kappa \textbf{u}_0=0$ in ${{\mathbb {R}}}^n$, which implies that $\textbf{u}_0$ is $\kappa$-homogeneous in ${{\mathbb {R}}}^n$. $\square$

Out next objective is the polynomial decay rate of the Weiss-type energy functional W at the regular free boundary points $x_0\in {\mathcal {R}}_\textbf{u}$, Lemma 9. It can be achieved with the help of the epiperimetric inequality, proved in [4]. To describe the inequality, we recall

$$\begin{aligned} M_{x_0}(\textbf{v})=\int _{B_1}\left( |\nabla \textbf{v}|^2+2F(x_0,\textbf{v})\right) -\kappa \int _{\partial B_1}|\textbf{v}|^2 \end{aligned}$$

and recall that ${{\mathbb {H}}}_{x_0}$ is a class of half-space solutions.

Theorem 7

(Epiperimetric inequality) There exist $\eta \in (0,1)$ and $\delta >0$ such that if $\textbf{c}\in W^{1,2}(B_1)$ is a homogeneous function of degree $\kappa$ and $\Vert \textbf{c}-\textbf{h}\Vert _{W^{1,2}(B_1)}\le \delta$ for some $\textbf{h}\in {{\mathbb {H}}}_{x_0}$, then there exists a function $\textbf{v}\in W^{1,2}(B_1)$ such that $\textbf{v}=\textbf{c}$ on $\partial B_1$ and $M_{x_0}(\textbf{v})\le (1-\eta )M_{x_0}(\textbf{c})+\eta M_{x_0}(\textbf{h})$.

For $x_0\in B_{1/2}$ and $0<r<1/2$, we denote the $\kappa$-homogeneous replacement of $\textbf{u}$ in $B_r(x_0)$ (or equivalently, the $\kappa$-homogeneous replacement of $\textbf{u}_{x_0,r}$ in $B_1$) by

$$\begin{aligned} \textbf{c}_{x_0,r}(x):=|x|^\kappa \textbf{u}_{x_0,r}\left( \frac{x}{|x|}\right) =\left( \frac{|x|}{r}\right) ^\kappa \textbf{u}\left( x_0+\frac{r}{|x|}x\right) ,\quad x\in {{\mathbb {R}}}^n. \end{aligned}$$

Lemma 9

Let $\textbf{u}$ be an almost minimizer in $B_1$ and $x_0\in {\mathcal {R}}_\textbf{u}\cap B_{1/2}$. Suppose that the epiperimetric inequality holds with $\eta \in (0,1)$ for each $\textbf{c}_{x_0,r},$ $0<r<r_1<1$. Then

$$\begin{aligned} W(\textbf{u},x_0,r)-W(\textbf{u},x_0,0+)\le Cr^\delta ,\quad 0<r<r_0 \end{aligned}$$

for some $\delta =\delta (n,\alpha ,\kappa ,\eta )>0$.

Proof

For simplicity we may assume $x_0=0$ and write $\textbf{u}_r=\textbf{u}_{0,r}$, $\textbf{c}_r=\textbf{c}_{0,r}$. For $W(\textbf{u},r)=W(\textbf{u},0,r)$, we define

$$\begin{aligned} e(r)&:=W(\textbf{u},r)-W(\textbf{u},0+)\\&=\frac{e^{ar^\alpha }}{r^{n+2\kappa -2}}\int _{B_r}\left( |\nabla \textbf{u}|^2+2F(0,\textbf{u})\right) -\frac{\kappa (1-br^\alpha )e^{ar^\alpha }}{r^{n+2\kappa -1}}\int _{\partial B_r}|\textbf{u}|^2-W(\textbf{u},0+), \end{aligned}$$

and compute

$$\begin{aligned} \frac{d}{dr}\left( \frac{e^{ar^\alpha }}{r^{n+2\kappa -2}}\right) =-\frac{(n+2\kappa -2)(1-Mr^\alpha )e^{ar^\alpha }}{r^{n+2\kappa -1}} \end{aligned}$$

and

$$\begin{aligned} \frac{d}{dr}\left( \frac{\kappa (1-br^\alpha )e^{ar^\alpha }}{r^{n+2\kappa -1}}\right) =\frac{-\kappa (1-br^\alpha )e^{ar^\alpha }(n+2\kappa -1+O(r^\alpha ))}{r^{n+2\kappa }}. \end{aligned}$$

Then

$$\begin{aligned} e'(r)&=-\frac{(n+2\kappa -2)(1-Mr^\alpha )e^{ar^\alpha }}{r^{n+2\kappa -1}}\int _{B_r}\left( |\nabla \textbf{u}|^2+2F(0,\textbf{u})\right) \\&\qquad +\frac{e^{ar^\alpha }}{r^{n+2\kappa -2}}\int _{\partial B_r}\left( |\nabla \textbf{u}|^2+2F(0,\textbf{u})\right) \\&\qquad +\frac{\kappa (1-br^\alpha )e^{ar^\alpha }(n+2\kappa -1+O(r^\alpha ))}{r^{n+2\kappa }}\int _{\partial B_r}|\textbf{u}|^2\\&\qquad -\frac{\kappa (1-br^\alpha )e^{ar^\alpha }}{r^{n+2\kappa -1}}\int _{\partial B_r}\left( 2\textbf{u}\partial _\nu \textbf{u}+\frac{n-1}{r}|\textbf{u}|^2\right) \\&\quad \ge -\frac{n+2\kappa -2}{r}\left( e(r)+\frac{\kappa (1-br^\alpha )e^{ar^\alpha }}{r^{n+2\kappa -1}}\int _{\partial B_r}|\textbf{u}|^2+W(\textbf{u},0+)\right) \\&\qquad +\frac{e^{ar^\alpha }}{r}\bigg [\frac{1}{r^{n+2\kappa -3}}\int _{\partial B_r}\left( |\nabla \textbf{u}|^2+2F(0,\textbf{u})\right) +\frac{2\kappa ^2+O(r^\alpha )}{r^{n+2\kappa -1}}\int _{\partial B_r}|\textbf{u}|^2\\&\qquad -\frac{2\kappa (1-br^\alpha )}{r^{n+2\kappa -2}}\int _{\partial B_r}\textbf{u}\partial _\nu \textbf{u}\bigg ]\\&\quad \ge -\frac{n+2\kappa -2}{r}\left( e(r)+W(\textbf{u},0+)\right) \\&\qquad +\frac{(1-br^\alpha )e^{ar^\alpha }}{r}\bigg [\frac{1}{r^{n+2\kappa -3}}\int _{\partial B_r}\left( |\nabla \textbf{u}|^2+2F(0,\textbf{u})\right) \\&\qquad +\frac{\kappa (2-n)+O(r^\alpha )}{r^{n+2\kappa -1}}\int _{\partial B_r}|\textbf{u}|^2-\frac{2\kappa }{r^{n+2\kappa -2}}\int _{\partial B_r}\textbf{u}\partial _\nu \textbf{u}\bigg ]. \end{aligned}$$

To simplify the last term, we observe that $\textbf{u}_r=\textbf{c}_r$ and $\partial _\nu \textbf{c}_r=\kappa \textbf{c}_r$ on $\partial B_1$ and that $|\nabla \textbf{c}_r|^2+2F(0,\textbf{c}_r)$ is homogeneous of degree $2\kappa -2$, and obtain

$$\begin{aligned}&\int _{\partial B_r}\left( \frac{1}{r^{n+2\kappa -3}}\left( |\nabla \textbf{u}|^2+2F(0,\textbf{u})\right) +\frac{\kappa (2-n)+O(r^\alpha )}{r^{n+2\kappa -1}}|\textbf{u}|^2-\frac{2\kappa }{r^{n+2\kappa -2}}\textbf{u}\partial _\nu \textbf{u}\right) \\&\quad =\int _{\partial B_1}\left( |\nabla \textbf{u}_r|^2+2F(0,\textbf{u}_r)+(\kappa (2-n)+O(r^\alpha ))|\textbf{u}_r|^2-2\kappa \textbf{u}_r\partial _\nu \textbf{u}_r\right) \\&\quad =\int _{\partial B_1}\left( |\partial _\nu \textbf{u}_r-\kappa \textbf{u}_r|^2+|\partial _\theta \textbf{u}_r|^2+2F(0,\textbf{u}_r)-\left( \kappa (n+\kappa -2)+O(r^\alpha )\right) |\textbf{u}_r|^2\right) \\&\quad \ge \int _{\partial B_1}\left( |\partial _\theta \textbf{c}_r|^2+2F(0,\textbf{c}_r)-\left( \kappa (n+\kappa -2)+O(r^\alpha )\right) |\textbf{c}_r|^2\right) \\&\quad =\int _{\partial B_1}\left( |\nabla \textbf{c}_r|^2+2F(0,\textbf{c}_r)-\left( \kappa (n+2\kappa -2)+O(r^\alpha )\right) |\textbf{c}_r|^2\right) \\&\quad =(n+2\kappa -2)\left[ \int _{B_1}\left( |\nabla \textbf{c}_r|^2+2F(0,\textbf{c}_r)\right) -(\kappa +O(r^\alpha ))\int _{\partial B_1}|\textbf{c}_r|^2\right] \\&\quad =(n+2\kappa -2)M_{0}(\textbf{c}_r)+O(r^\alpha )\int _{\partial B_1}|\textbf{u}_r|^2. \end{aligned}$$

Thus

$$\begin{aligned} \begin{aligned} e'(r)&\ge -\frac{n+2\kappa -2}{r}\left( e(r)+W(\textbf{u},0+)\right) +\frac{(1-br^\alpha )e^{ar^\alpha }(n+2\kappa -2)}{r}M_{0}(\textbf{c}_r)\\&\quad +\frac{O(r^\alpha )}{r}\int _{\partial B_1}|\textbf{u}_r|^2. \end{aligned} \end{aligned}$$

(6.2)

We want to estimate $M_0(\textbf{c}_r)$. From the assumption that the epiperimetric inequality holds for $\textbf{c}_r$, we have $M_{0}(\textbf{v}^r)\le (1-\eta )M_{0}(\textbf{c}_r)+\eta M_{0}(\textbf{h}^r)$ for some $\textbf{h}^r\in {{\mathbb {H}}}_0$ and $\textbf{v}^r\in W^{1,2}(B_1)$ with $\textbf{v}^r=\textbf{c}_r$ on $\partial B_1$. In addition, $0\in {\mathcal {R}}_\textbf{u}$ ensures that there exists a sequence $t_j\rightarrow 0+$ such that $\textbf{u}_{t_j}\rightarrow \textbf{h}_0$ in $C^1_{{\text {loc}}}({{\mathbb {R}}}^n)$ for some $\textbf{h}_0\in {{\mathbb {H}}}_0$. Then

$$\begin{aligned} W(\textbf{u},0+)&=\lim _{j\rightarrow \infty }W(\textbf{u},t_j)\\&=\lim _{j\rightarrow \infty }e^{at_j^\alpha }\left[ \int _{B_1}\left( |\nabla \textbf{u}_{t_j}|^2+2F(0,\textbf{u}_{t_j})\right) -\kappa (1-bt_j^\alpha )\int _{\partial B_1}|\textbf{u}_{t_j}|^2\right] \\&=M_{0}(\textbf{h}_{0})=M_{0}(\textbf{h}^r). \end{aligned}$$

Here, the last equality holds since both $\textbf{h}_j$ and $\textbf{h}^r$ are elements in ${{\mathbb {H}}}_0$. By the epiperimetric inequality and the almost-minimality of $\textbf{u}_r$,

$$\begin{aligned} (1-\eta )M_{0}(\textbf{c}_r)+\eta W(\textbf{u},0+)&\ge M_{0}(\textbf{v}^r)=\int _{B_1}\left( |\nabla \textbf{v}^r|^2+2F(0,\textbf{v}^r)\right) -\kappa \int _{\partial B_1}|\textbf{v}^r|^2\\&\ge \frac{1}{1+Mr^\alpha }\int _{B_1}\left( |\nabla \textbf{u}_r|^2+2F(0,\textbf{u}_r)\right) -\kappa \int _{\partial B_1}|\textbf{u}_r|^2\\&=\frac{e^{-ar^\alpha }}{1+Mr^\alpha }W(\textbf{u},r)+O(r^\alpha )\int _{\partial B_1}|\textbf{u}_r|^2. \end{aligned}$$

We rewrite it as

$$\begin{aligned} M_{0}(\textbf{c}_r)\ge \frac{\frac{e^{-ar^\alpha }}{1+Mr^\alpha }W(\textbf{u},r)-\eta W(\textbf{u},0+)}{1-\eta }+O(r^\alpha )\int _{\partial B_1}|\textbf{u}_r|^2 \end{aligned}$$

and, combining this with (6.2), obtain

$$\begin{aligned} e'(r)&\ge -\frac{n+2\kappa -2}{r}\left( e(r)+W(\textbf{u},0+)\right) \\&\quad +\frac{(1-br^\alpha )e^{ar^\alpha }(n+2\kappa -2)}{r}\left( \frac{\frac{e^{-ar^\alpha }}{1+Mr^\alpha }W(\textbf{u},r)-\eta W(\textbf{u},0+)}{1-\eta }\right) \\&\quad +\frac{O(r^\alpha )}{r}\int _{\partial B_1}|\textbf{u}_r|^2. \end{aligned}$$

Note that from Theorem 2, there is a constant $C>0$ such that

$$\begin{aligned} W(\textbf{u},0+)\le W(\textbf{u},r)\le C \end{aligned}$$

and

$$\begin{aligned} \int _{\partial B_1}|\textbf{u}_r|^2=\frac{1}{r^{n+2\kappa -1}}\int _{\partial B_r}|\textbf{u}|^2\le C \end{aligned}$$

for small $r>0$. Then

$$\begin{aligned} e'(r)&\ge -\frac{n+2\kappa -2}{r}\left( e(r)+W(\textbf{u},0+)\right) +\frac{n+2\kappa -2}{r}\left( \frac{W(\textbf{u},r)-\eta W(\textbf{u},0+)}{1-\eta }\right) \\&\quad +\frac{O(r^\alpha )}{r}\left( W(\textbf{u},r)+W(\textbf{u},0+)+\int _{\partial B_1}|\textbf{u}_r|^2\right) \\&\ge -\frac{n+2\kappa -2}{r}e(r)+\frac{n+2\kappa -2}{r}\left( \frac{W(\textbf{u},r)-W(\textbf{u},0+)}{1-\eta }\right) -C_1r^{\alpha -1}\\&=\left( \frac{(n+2\kappa -2)\eta }{1-\eta }\right) \frac{e(r)}{r}-C_1r^{\alpha -1}. \end{aligned}$$

Now, take $\delta =\delta (n,\alpha ,\kappa ,\eta )$ such that $0<\delta <\min \left\{ \frac{(n+2\kappa -2)\eta }{1-\eta },\alpha \right\}$. Using the differential inequality above for e(r) and that $e(r)=W(\textbf{u},r)-W(\textbf{u},0+)\ge 0$, we have for $0<r<r_0$

$$\begin{aligned} \frac{d}{dr}\left[ e(r)r^{-\delta }+\frac{C_1}{\alpha -\delta }r^{\alpha -\delta }\right]&=r^{-\delta }\left[ e'(r)-\frac{\delta }{r}e(r)\right] +C_1r^{\alpha -\delta -1}\\&\ge r^{-\delta }\left[ \left( \frac{(n+2\kappa -2)\eta }{1-\eta }-\delta \right) \frac{e(r)}{r}-C_1r^{\alpha -1}\right] +C_1r^{\alpha -\delta -1}\\&\ge 0. \end{aligned}$$

Thus

$$\begin{aligned} e(r)r^{-\delta }\le e(r_0)r_0^{-\delta }+\frac{C_0}{\alpha -\delta }r_0^{\alpha -\delta }, \end{aligned}$$

and hence we conclude that

$$\begin{aligned} W(\textbf{u},r)-W(\textbf{u},0+)=e(r)\le Cr^\delta . \end{aligned}$$

$\square$

Now, we consider an auxiliary function

$$\begin{aligned} \phi (r):=e^{-(\kappa b/\alpha )r^\alpha }r^{\kappa },\quad r>0, \end{aligned}$$

which is a solution of the differential equation

$$\begin{aligned} \phi '(r)=\kappa \,\phi (r)\frac{1-b r^\alpha }{r},\quad r>0. \end{aligned}$$

For $x_0\in B_{1/2}$, we define the $\kappa$-almost homogeneous rescalings by

$$\begin{aligned} \textbf{u}_{x_0, r}^{\phi }(x):=\frac{\textbf{u}(rx+x_0)}{\phi (r)},\quad x\in B_{1/(2r)}. \end{aligned}$$

Lemma 10

(Rotation estimate) Under the same assumption as in Lemma 9,

$$\begin{aligned} \int _{\partial B_1}|\textbf{u}_{x_0,t}^\phi -\textbf{u}_{x_0,s}^\phi |\le Ct^{\delta /2},\quad s<t<t_0. \end{aligned}$$

Proof

Without loss of generality, assume $x_0=0$. For $\textbf{u}_r^\phi =\textbf{u}_{0,r}^\phi$ and $W(\textbf{u},r)=W(\textbf{u},0,r)$,

$$\begin{aligned} \frac{d}{dr}\textbf{u}_{r}^{\phi }(x)&=\frac{\nabla \textbf{u}(rx)\cdot x}{\phi (r)}-\frac{\textbf{u}(rx)[\phi '(r)/\phi (r)]}{\phi (r)}=\frac{1}{\phi (r)}\left( \nabla \textbf{u}(rx)\cdot x-\frac{\kappa (1-br^\alpha )}{r}\textbf{u}(rx)\right) . \end{aligned}$$

By Theorem 5, we have for $0<r<t_0$

$$\begin{aligned}&\left( \int _{\partial B_1}\left[ \frac{d}{dr}\textbf{u}_r^\phi (\xi )\right] ^2dS_\xi \right) ^{1/2}\\&= \frac{1}{\phi (r)}\left( \int _{\partial B_1}\left| \nabla \textbf{u}(r\xi )\cdot \xi -\frac{\kappa (1-br^\alpha )}{r}\textbf{u}(r \xi )\right| ^2 dS_\xi \right) ^{1/2}\\&=\frac{1}{\phi (r)}\left( \frac{1}{r^{n-1}}\int _{\partial B_r}\left| \nabla \textbf{u}(x)\cdot \nu -\frac{\kappa (1-br^\alpha )}{r} \textbf{u}(x)\right| ^2dS_x\right) ^{1/2}\\&\le \frac{1}{\phi (r)}\left( \frac{1}{r^{n-1}}\frac{r^{n+2\kappa -2}}{e^{a r^\alpha }}\frac{d}{dr}W(\textbf{u},r)\right) ^{1/2}=\frac{e^{c r^\alpha }}{r^{1/2}}\left( \frac{d}{dr}W(\textbf{u},r)\right) ^{1/2} ,\quad c=\frac{\kappa b}{\alpha }-\frac{a}{2}. \end{aligned}$$

Using this and Lemma 9, we can compute

$$\begin{aligned} \int _{\partial B_1}|\textbf{u}_{t}^\phi -\textbf{u}_{s}^\phi |&\le \int _{\partial B_1}\int _s^t\left| \frac{d}{dr}\textbf{u}_{r}^\phi \right| \,dr=\int _s^t\int _{\partial B_1}\left| \frac{d}{dr}\textbf{u}_{r}^\phi \right| \,dr\\&\le C_n\int _s^t\left( \int _{\partial B_1}\left| \frac{d}{dr}\textbf{u}_{r}^\phi \right| ^2\right) ^{1/2}\,dr\\&\le C_n\left( \int _s^t\frac{1}{r}\,dr\right) ^{1/2}\left( \int _s^tr\int _{\partial B_1}\left| \frac{d}{dr}\textbf{u}_{r}^\phi \right| ^2\,dr\right) ^{1/2}\\&\le C_ne^{ct^\alpha }\left( \log \frac{t}{s}\right) ^{1/2}\left( \int _s^t\frac{d}{dr}W(\textbf{u},r)\,dr\right) ^{1/2}\\&\le C\left( \log \frac{t}{s}\right) ^{1/2}(W(\textbf{u},t)-W(\textbf{u},s))^{1/2}\\&\le C\left( \log \frac{t}{s}\right) ^{1/2}t^{\delta /2},\quad 0<t<t_0. \end{aligned}$$

Now, by a standard dyadic argument, we conclude that

$$\begin{aligned} \int _{\partial B_1}|\textbf{u}_{t}^\phi -\textbf{u}_{s}^\phi |\le Ct^{\delta /2}. \end{aligned}$$

$\square$

The following are generalization of Lemma 4.4 and Proposition 4.6 in [4] on $\kappa$-homogeneous solutions from the case $\lambda _{\pm }=1$ to the constant coefficients $\lambda _\pm (x_0)$, which can be proved in similar fashion.

Lemma 11

For every $x_0\in B_{1/2}$, ${{\mathbb {H}}}_{x_0}$ is isolated (in the topology of $W^{1,2}(B_1)$) within the class of $\kappa$-homogeneous solutions $\textbf{v}$ for $\Delta \textbf{v}=f(x_0,\textbf{v})$.

Proposition 2

For $x_0\in B_{1/2}$, let $\textbf{v}\not \equiv 0$ be a $\kappa$-homogeneous solution of $\Delta \textbf{v}=f(x_0,\textbf{v})$ satisfying $\{|\textbf{v}|=0\}^{\textrm{0}}\ne \emptyset$. Then $M_{x_0}(\textbf{v})\ge {\mathcal {B}}_{x_0}$ and equality implies $\textbf{v}\in {{\mathbb {H}}}_{x_0}$; here ${\mathcal {B}}_{x_0}=M_{x_0}(\textbf{h})$ for every $\textbf{h}\in {{\mathbb {H}}}_{x_0}$.

The proof of the next proposition can be obtained as in Proposition 4.5 in [1], by using Lemma 11 and a continuity argument.

Proposition 3

If $x_0\in {\mathcal {R}}_\textbf{u}$, then all blowup limits of $\textbf{u}$ at $x_0$ belong to ${{\mathbb {H}}}_{x_0}$.

7 Regularity of the regular set

In this last section we establish one of the main result in this paper, the $C^{1,\gamma }$-regularity of the regular set ${\mathcal {R}}_\textbf{u}$.

We begin by showing that ${\mathcal {R}}_\textbf{u}$ is an open set in $\Gamma (\textbf{u})$.

Lemma 12

${\mathcal {R}}_\textbf{u}$ is open relative to $\Gamma (\textbf{u})$.

Proof

Step 1. For points $y\in B_{1/2}$, we let ${\mathbb {A}}_{y}$ be a set of $\kappa$-homogeneous solutions $\textbf{v}$ of $\Delta \textbf{v}=f(y,\textbf{v})$ satisfying $\textbf{v}\not \in {{\mathbb {H}}}_{y}$, and define

$$\begin{aligned} \rho _{y}:=\inf \{\Vert \textbf{h}-\textbf{v}\Vert _{C^{1}(\overline{B_1})}:\,\textbf{h}\in {{\mathbb {H}}}_{y},\textbf{v}\in {\mathbb {A}}_{y}\}. \end{aligned}$$

From $\Vert \textbf{h}-\textbf{v}\Vert _{C^1(\overline{B_1})}\ge c(n)\Vert \textbf{h}-\textbf{v}\Vert _{W^{1,2}(B_1)}$, the isolation property of ${{\mathbb {H}}}_{x_0}$ in Lemma 11 also holds in $C^1(\overline{B_1})$-norm, thus $\rho _y>0$ for every $y\in B_{1/2}$.

We claim that there is a universal constant $c_1>0$ such that $c_1<\rho _y<1/c_1$ for all $y\in B_{1/2}$. Indeed, the second inequality $\rho _y<1/c_1$ is obvious. For the first one, we assume to the contrary that $\rho _{y_i}\rightarrow 0$ for a sequence $y_i\in B_{1/2}$. This gives sequences $\textbf{h}_{y_i}\in {{\mathbb {H}}}_{y_i}$ and $\textbf{v}_{y_i}\in {\mathbb {A}}_{y_i}$ such that ${\text {dist}}(\textbf{h}_{y_i},\textbf{v}_{y_i})\rightarrow 0$, where the distance is measured in $C^1(\overline{B_1})$-norm. Over a subsequence, we have $y_i\rightarrow y_0$ and, using that $\textbf{h}_{y_i}$ and $\textbf{v}_{y_i}$ are uniformly bounded, $\textbf{h}_{y_i}\rightarrow \textbf{h}_{y_0}$ and $\textbf{v}_{y_i}\rightarrow \textbf{v}_{y_0}$ for some $\textbf{h}_{y_0}\in {{\mathbb {H}}}_{y_0}$ and $\textbf{v}_{y_0}\in {\mathbb {A}}_{y_0}$. It follows that ${\text {dist}}(\textbf{h}_{y_0},\textbf{v}_{y_0})=\lim _{i\rightarrow \infty }{\text {dist}}(\textbf{h}_{y_i},\textbf{v}_{y_i})=0$, meaning that $\textbf{h}_{y_0}=\textbf{v}_{y_0}$, a contradiction.

Step 2. Towards a contradiction we assume that the statement of Lemma 12 is not true. Then we can find a point $x_0\in {\mathcal {R}}_\textbf{u}$ and a sequence of points $x_i\in \Gamma (\textbf{u})\setminus {\mathcal {R}}_\textbf{u}$ such that $x_i\rightarrow x_0$.

For a small constant $\varepsilon _1=\varepsilon _1(n,q,M,c_1)>0$ to be specified later, we claim that there is a sequence $r_i\rightarrow 0$ and a subsequence of $x_i$, still denoted by $x_i$, such that

$$\begin{aligned} {\text {dist}}(\textbf{u}_{x_i,r_i},{{\mathbb {H}}}_{x_0})=\varepsilon _1, \end{aligned}$$

(7.1)

where $\textbf{u}_{x_i,r_i}(x)=\frac{\textbf{u}(x_i+r_ix)}{r_i^\kappa }$ are the $\kappa$-homogeneous rescalings.

Indeed, since $x_0\in {\mathcal {R}}_\textbf{u}$, we can find a sequence $t_j\rightarrow 0$ such that

$$\begin{aligned} {\text {dist}}(\textbf{u}_{x_0,t_j},{{\mathbb {H}}}_{x_0})<\varepsilon _1/2. \end{aligned}$$

(7.2)

For each fixed $t_j$, we take a point from the above sequence $\{x_i\}_{i=1}^\infty$ (and redefine the point as $x_j$ for convenience) close to $x_0$ such that

$$\begin{aligned} {\text {dist}}(\textbf{u}_{x_j,t_j},{{\mathbb {H}}}_{x_0})\le {\text {dist}}(\textbf{u}_{x_j,t_j},\textbf{u}_{x_0,t_j})+{\text {dist}}(\textbf{u}_{x_0,t_j},{{\mathbb {H}}}_{x_0})<\varepsilon _1. \end{aligned}$$

On the other hand, using $x_j\not \in {\mathcal {R}}_\textbf{u}$ and Proposition 3, we see that there is $\tau _j<t_j$ such that ${\text {dist}}(\textbf{u}_{x_j,\tau _j},{{\mathbb {H}}}_{x_j})>\rho _{x_j}/2$. Using the result in Step 1, we have ${\text {dist}}(\textbf{u}_{x_j,\tau _j},{{\mathbb {H}}}_{x_j})>c_1/2>2\varepsilon _1$ for small $\varepsilon _1$. We next want to show that for large j

$$\begin{aligned} {\text {dist}}(\textbf{u}_{x_j,\tau _j},{{\mathbb {H}}}_{x_0})\ge (3/2)\varepsilon _1. \end{aligned}$$

(7.3)

For this aim, we let $\textbf{h}_{x_0}\in {{\mathbb {H}}}_{x_0}$ be given. Then $\textbf{h}_{x_j}:=\frac{\beta _{x_j}}{\beta _{x_0}}\textbf{h}_{x_0}\in {{\mathbb {H}}}_{x_j}$ and $\frac{\beta _{x_j}}{\beta _{x_0}}=\left( \frac{\lambda _+(x_j)}{\lambda _+(x_0)}\right) ^{\kappa /2}\rightarrow 1$ as $j\rightarrow \infty$. Thus ${\text {dist}}(\textbf{h}_{x_j},\textbf{h}_{x_0})<\varepsilon _1/2$ for large j, and hence

$$\begin{aligned} {\text {dist}}(\textbf{u}_{x_j,\tau _j},\textbf{h}_{x_0})&\ge {\text {dist}}(\textbf{u}_{x_j,\tau _j},\textbf{h}_{x_j})-{\text {dist}}(\textbf{h}_{x_j},\textbf{h}_{x_0})\\&>2\varepsilon _1-\varepsilon _1/2=(3/2)\varepsilon _1. \end{aligned}$$

This implies (7.3).

Now, (7.2) and (7.3) ensure the existence of $r_j\in (\tau _j,t_j)$ such that

$$\begin{aligned} {\text {dist}}(\textbf{u}_{x_j,r_j},{{\mathbb {H}}}_{x_0})=\varepsilon _1. \end{aligned}$$

Step 3. (7.1) implies that $\{\textbf{u}_{x_i,r_i}\}$ is uniformly bounded in $C^1$-norm (so in $C^{1,\alpha /2}$-norm as well), thus we can follow the argument in Step 1–2 in the proof of Lemma 8 with $\textbf{u}_{x_i,r_i}$ in the place of $\textbf{u}_t$ to have that over a subsequence

$$\begin{aligned} \textbf{u}_{x_i,r_i}\rightarrow \textbf{u}^*\quad \text {in }C^1_{{\text {loc}}}({{\mathbb {R}}}^n) \end{aligned}$$

for some nonzero global solution $\textbf{u}^*$ of $\Delta \textbf{u}^*=f(x_0,\textbf{u}^*)$.

From (7.1) again, we may assume that $\Vert \textbf{u}_{x_i,r_i}-\textbf{h}\Vert _{C^1(\overline{B_1})}\le 2\varepsilon _1$ for $\textbf{h}(x)=\beta _{x_0}(x^1_+)^\kappa$. Then $|\textbf{u}_{x_i,r_i}|+|\nabla \textbf{u}_{x_i,r_i}|\le 2\varepsilon _1$ in $B_1\cap \{x_1\le 0\}$. By the nondegeneracy property, Lemma 7, we know that $\int _{B_t(z)}\left( |\nabla \textbf{u}_{x_i,r_i}|^2+|\textbf{u}_{x_i,r_i}|^{q+1}\right) \ge \varepsilon _0t^{n+2\kappa -2}$ for any $z\in \overline{\{|\textbf{u}_{x_i,r_i}|>0\}}$ and $B_t(z)\Subset B_1$. We can deduce from the preceding two inequalities that if $\varepsilon _1$ is small, then $\textbf{u}_{x_i,r_i}\equiv 0$ in $B_1\cap \{x_1\le -1/4\}$. Therefore, the coincidence set $\{|\textbf{u}^*|=0\}$ has a nonempty interior, and there exist an open ball $D\Subset \{|\textbf{u}^*|=0\}$ and a point $z_0\in \partial D\cap \partial \{|\textbf{u}^*|>0\}$.

Step 4. We claim that $\Vert \textbf{u}^*\Vert _{L^\infty (B_r(z_0))}=O(r^\kappa )$.

In fact, we can proceed as in the proof of the sufficiency part of Theorem 1.2 in [4]. In the theorem, they assume $\Vert \textbf{u}\Vert _{L^\infty (B_r)}=o(r^{\lfloor \kappa \rfloor })$ and prove $\Vert \textbf{u}\Vert _{L^\infty (B_r)}=O(r^\kappa )$. We want to show that the condition $\Vert \textbf{u}\Vert _{L^\infty (B_r)}=o(r^{\lfloor \kappa \rfloor })$ can be replaced by $0\in \partial D\cap \partial \{|\textbf{u}|>0\}$, where D is an open ball contained in $\{|\textbf{u}|=0\}$ (then it can be applied to $\textbf{u}^*$ in our case, and $\Vert \textbf{u}^*\Vert _{L^\infty (B_r(z_0))}=O(r^\kappa )$ follows). Indeed, the growth condition on $\textbf{u}$ in the theorem is used only to prove the following: if ${\tilde{\textbf{u}}}_j(x)=P_j(x)+\Gamma _j(x)$ in $B_r$, where ${\tilde{\textbf{u}}}_j(x)=\frac{\textbf{u}(rx)}{jr_j^\kappa }$ is a rescaling of $\textbf{u}$ at 0, $P_j$ is a harmonic polynomial of degree $l\le \lfloor \kappa \rfloor$ and $|\Gamma _j(x)|\le C|x|^{l+\varepsilon }$, $0<\varepsilon <1$, then $P_j\equiv 0$. To see that $0\in \partial D\cap \partial \{|\textbf{u}|>0\}$ also gives the same result, we observe that it implies that ${\tilde{\textbf{u}}}_j=0$ in an open subset $A_r$ of the ball $B_r$. This is possible only when $P_j=\Gamma _j=0$ in $A_r$. Thus, by the unique continuation $P_j\equiv 0$.

Step 5. Recall the standard Weiss energy

$$\begin{aligned} W^0(\textbf{u}^*,z,y,s)=\frac{1}{s^{n+2\kappa -2}}\int _{B_s(z)}\left( |\nabla \textbf{u}^*|^2+2F(y,\textbf{u}^*)\right) -\frac{\kappa }{s^{n+2\kappa -1}}\int _{\partial B_s(z)}|\textbf{u}^*|^2. \end{aligned}$$

By the result in Step 4, there exists a $\kappa$-homogeneous blowup $\textbf{u}^{**}$ of $\textbf{u}^*$ at $z_0$ (i.e., $\textbf{u}^{**}(x)=\lim _{r\rightarrow 0}\frac{\textbf{u}^*(rx+z_0)}{r^\kappa }$ over a subsequence). From $z_0\in \partial D$, $\textbf{u}^{**}$ should have a nonempty interior of the zero-set $\{|\textbf{u}^{**}|=0\}$ near the origin 0, thus by Proposition 2

$$\begin{aligned} W^0(\textbf{u}^*,z_0,x_0,0+)=M_{x_0}(\textbf{u}^{**})\ge {\mathcal {B}}_{x_0}. \end{aligned}$$

Moreover, we observe that for any $\rho \in (0,1)$ and $s\in (0,1)$

$$\begin{aligned} W^0(\textbf{u}^*,z_0,x_0,0+)&\le W^0(\textbf{u}^*,z_0,x_0,\rho )\\&=\lim _{i\rightarrow \infty }W^0(\textbf{u}_{x_i,r_i},z_0,x_i+r_iz_0,\rho )\\&=\lim _{i\rightarrow \infty }W^0(\textbf{u},x_i+r_iz_0,x_i+r_iz_0,\rho r_i)\\&\le \lim _{i\rightarrow \infty }W(\textbf{u},x_i+r_iz_0,\rho r_i)\\&\le \lim _{i\rightarrow \infty }W(\textbf{u},x_i+r_iz_0,s)=W(\textbf{u},x_0,s). \end{aligned}$$

Here, the first inequality follows from $\Delta \textbf{u}^*=f(x_0,\textbf{u}^*)$ and Weiss monotonicity formula. We have used in the second step that $|\nabla \textbf{u}_{x_i,r_i}|^2+2F(x_i+r_iz_0,\textbf{u}_{x_i,r_i})$ and $|\textbf{u}_{x_i,r_i}|^2$ converge uniformly to $|\nabla \textbf{u}^*|^2+2F(x_0,\textbf{u}^*)$ and $|\textbf{u}^*|^2$, respectively. The fourth step follows from the definition of $W^0$ and W together with the fact that $W^0(\textbf{u},x_i+r_iz_0,x_i+r_iz_0,\rho r_i)$ is positive for large i (since $\lim _{i\rightarrow \infty }W^0(\textbf{u},x_i+r_iz_0,x_i+r_iz_0,\rho r_i)\ge {\mathcal {B}}_{x_0}$). Next, taking $s\searrow 0$ and using $W(\textbf{u},x_0,0+)={\mathcal {B}}_{x_0}$, we obtain that $W^0(\textbf{u}^*,z_0,x_0,0+)\le {\mathcal {B}}_{x_0}$. Thus we conclude $W^0(\textbf{u}^*,z_0,x_0,\rho )={\mathcal {B}}_{x_0}$ for $0<\rho <1$, and hence $\textbf{u}^*$ is a $\kappa$-homogeneous function with respect to $z_0$.

Now, we can apply Proposition 2 to obtain that $\textbf{u}^*$ is a half-space solution with respect to $z_0$, i.e., $\textbf{u}^*(\cdot -z_0)\in {{\mathbb {H}}}_{x_0}$. Since $\textbf{u}_{x_i,r_i}$ satisfies $|\textbf{u}_{x_i,r_i}(0)|=0$ and the nondegeneracy $\int _{B_t}\left( |\nabla \textbf{u}_{x_i,r_i}|^2+|\textbf{u}_{x_i,r_i}|^{q+1}\right) \ge \varepsilon _0 t^{n+2\kappa -2}$, $\textbf{u}^*$ also satisfies the similar equations, and thus $z_0=0$. This implies $\textbf{u}^*\in {{\mathbb {H}}}_{x_0}$, which contradicts (7.1). $\square$

Lemma 13

Let $C_h$ be a compact subset of ${\mathcal {R}}_\textbf{u}$. For any $\varepsilon >0$, there is $r_0>0$ such that if $x_0\in C_h$ and $0<r<r_0$, then the $\kappa$-homogeneous replacement $\textbf{c}_{x_0,r}$ of $\textbf{u}$ satisfies

$$\begin{aligned} {\text {dist}}(\textbf{c}_{x_0,r},{{\mathbb {H}}}_{x_0})<\varepsilon , \end{aligned}$$

(7.4)

where the distance is measured in the $C^1(\overline{B_1})$-norm.

Proof

We claim that for any $\varepsilon >0$, there is $r_0>0$ such that

$$\begin{aligned} {\text {dist}}(\textbf{u}_{x_0,r},{{\mathbb {H}}}_{x_0})<\varepsilon \quad \text {for any }x_0\in C_h\text { and }0<r<r_0, \end{aligned}$$

which readily gives (7.4) (see the proof of Lemma 10 in [2]).

Towards a contradiction we assume that there exist a constant $\varepsilon _0>0$ and sequences $x_j\in C_h$ (converging to $x_0\in C_h$) and $r_j\rightarrow 0$ such that

$$\begin{aligned} {\text {dist}}(\textbf{u}_{x_j,r_j},{{\mathbb {H}}}_{x_j})\ge \varepsilon _0. \end{aligned}$$

By a continuity argument, for each $\theta \in (0,1)$ we can find a sequence $t_j<r_j$ such that

$$\begin{aligned} {\text {dist}}(\textbf{u}_{x_j,t_j},{{\mathbb {H}}}_{x_j})=\theta \varepsilon _0. \end{aligned}$$

By following the argument in Step 1–2 in the proof of Lemma 8 with $\textbf{u}_{x_j,t_j}$ in the place of $\textbf{u}_t$, we can show that up to a subsequence

$$\begin{aligned} \textbf{u}_{x_j,t_j}\rightarrow \textbf{u}_{x_0}\quad \text {in }C^1_{{\text {loc}}}({{\mathbb {R}}}^n;{{\mathbb {R}}}^m) \end{aligned}$$

for some nonzero global solution $\textbf{u}_{x_0}\in C^1_{{\text {loc}}}({{\mathbb {R}}}^n;{{\mathbb {R}}}^m)$ of $\Delta \textbf{u}_{x_0}=f(x_0,\textbf{u}_{x_0})$. We remark that the blowup $\textbf{u}_{x_0}$ depends on the sequence $\{t_j\}$, thus on the choice of $0<\theta <1$.

From ${\text {dist}}(\textbf{u}_{x_j,t_j},{{\mathbb {H}}}_{x_j})=\theta \varepsilon _0$, we can take $\textbf{h}_{x_j}\in {{\mathbb {H}}}_{x_j}$ such that ${\text {dist}}(\textbf{u}_{x_j,t_j},\textbf{h}_{x_j})\le 2\theta \varepsilon _0.$ For each j we define $\textbf{h}_{x_0}^j:=\frac{\beta _{x_0}}{\beta _{x_j}}\textbf{h}_{x_j}\in {{\mathbb {H}}}_{x_0}$. Since $\frac{\beta _{x_0}}{\beta _{x_j}}=\left( \frac{\lambda _+(x_0)}{\lambda _+(x_j)}\right) ^{\kappa /2}\rightarrow 1$,

$$\begin{aligned} {\text {dist}}(\textbf{h}_{x_j},\textbf{h}_{x_0}^j)\le o(|x_j-x_0|). \end{aligned}$$

Thus,

$$\begin{aligned} {\text {dist}}(\textbf{u}_{x_0},\textbf{h}_{x_0}^j)&\le {\text {dist}}(\textbf{u}_{x_0},\textbf{u}_{x_j,t_j})+{\text {dist}}(\textbf{u}_{x_j,t_j},\textbf{h}_{x_j})+{\text {dist}}(\textbf{h}_{x_j},\textbf{h}_{x_0}^j)\\&\le 2\theta \varepsilon _0+o(|x_j-x_0|), \end{aligned}$$

and hence

$$\begin{aligned} {\text {dist}}(\textbf{u}_{x_0},{{\mathbb {H}}}_{x_0})\le \limsup _{j\rightarrow \infty }{\text {dist}}(\textbf{u}_{x_0},\textbf{h}_{x_0}^j)\le 2\theta \varepsilon _0. \end{aligned}$$

On the other hand, for each $h^{x_0}\in {{\mathbb {H}}}_{x_0}$ we let $\textbf{h}^{x_j}:=\frac{\beta _{x_j}}{\beta _{x_0}}\textbf{h}^{x_0}\in {{\mathbb {H}}}_{x_j}$ so that ${\text {dist}}(\textbf{h}^{x_j},\textbf{h}^{x_0})\le o(|x_j-x_0|)$. Using ${\text {dist}}(\textbf{u}_{x_j,t_j},{{\mathbb {H}}}_{x_j})=\theta \varepsilon _0$ again, we obtain

$$\begin{aligned} {\text {dist}}(\textbf{u}_{x_0},\textbf{h}^{x^0})&\ge {\text {dist}}(\textbf{u}_{x_j,t_j},\textbf{h}^{x_j})-{\text {dist}}(\textbf{h}^{x_j},\textbf{h}^{x_0})-{\text {dist}}(\textbf{u}_{x_j,t_j},\textbf{u}_{x_0})\\&\ge \theta \varepsilon _0-o(|x_j-x_0|), \end{aligned}$$

and conclude

$$\begin{aligned} {\text {dist}}(\textbf{u}_{x_0},{{\mathbb {H}}}_{x_0})\ge \theta \varepsilon _0. \end{aligned}$$

Therefore,

$$\begin{aligned} \theta \varepsilon _0\le {\text {dist}}(\textbf{u}_{x_0},{{\mathbb {H}}}_{x_0})\le 2\theta \varepsilon _0. \end{aligned}$$

For $\theta >0$ small enough, this inequality contradicts the isolation property of ${{\mathbb {H}}}_{x_0}$ in Lemma 11, provided $\textbf{u}_{x_0}$ is homogeneous of degree $\kappa$.

To prove the homogeneity, we fix $0<r<R<\infty$ and follow the argument in Step 3 in Lemma 8 to obtain

$$\begin{aligned} \begin{aligned}&W(\textbf{u},x_j,Rt)-W(\textbf{u},x_j,rt)\\&\quad \ge \int _r^R\frac{1}{\sigma ^{n+2\kappa }}\int _{\partial B_\sigma }\left| x\cdot \nabla \textbf{u}_{x_j,t}-\kappa \left( 1-b(t\sigma )^\alpha \right) \textbf{u}_{x_j,t}\right| ^2\,dS_xd\sigma \end{aligned} \end{aligned}$$

(7.5)

for small t. Recall the standard Weiss energies

$$\begin{aligned} W^0(\textbf{u},x_j,t)=\frac{1}{t^{n+2\kappa -2}}\int _{B_t(x_j)}\left( |\nabla \textbf{u}|^2+2F(x_j,\textbf{u})\right) -\frac{\kappa }{t^{n+2\kappa -1}}\int _{\partial B_t(x_j)}|\textbf{u}|^2. \end{aligned}$$

We have

$$\begin{aligned} M_{x_j}(\textbf{u}_{x_j,t})=W^0(\textbf{u},x_j,t)=W(\textbf{u},x_j,t)+O(t^\alpha ), \end{aligned}$$

where the second equality holds by Theorem 2. This, together with $x_j\in {{\mathcal {R}}}_\textbf{u}$ and the monotonicity of $W(\textbf{u},x_j,\cdot )$, gives that

$$\begin{aligned} W(\textbf{u},x_j,t)\searrow {{\mathcal {B}}}_{x_j}\quad \text {as }t\searrow 0. \end{aligned}$$

Applying Dini’s theorem gives that for any $\varepsilon >0$ there exists $t_0=t_0(\varepsilon )>0$ such that

$$\begin{aligned} {{\mathcal {B}}}_{x_j}\le W(\textbf{u},x_j,t)\le {{\mathcal {B}}}_{x_j}+\varepsilon \quad \text {for any }t<t_0\text { and }x_j\in C_h. \end{aligned}$$

Then, for large j, we have ${{\mathcal {B}}}_{x_j}\le W(\textbf{u},x_j,Rt_j)\le {{\mathcal {B}}}_{x_j}+\varepsilon$, and thus

$$\begin{aligned} {{\mathcal {B}}}_{x_0}\le \liminf _{j\rightarrow \infty }W(\textbf{u},x_j,Rt_j)\le \limsup _{j\rightarrow \infty }W(\textbf{u},x_j,Rt_j)\le {{\mathcal {B}}}_{x_0}+\varepsilon . \end{aligned}$$

Taking $\varepsilon \searrow 0$, we obtain that

$$\begin{aligned} \lim _{j\rightarrow \infty }W(\textbf{u},x_j,Rt_j)={{\mathcal {B}}}_{x_0}. \end{aligned}$$

Similarly, we have $\lim _{j\rightarrow \infty }W(\textbf{u},x_j,rt_j)={{\mathcal {B}}}_{x_0}$. They enable us to take $j\rightarrow \infty$ in (7.5) to get

$$\begin{aligned} 0=\int _r^R\frac{1}{\sigma ^{n+2\kappa }}\int _{\partial B_\sigma }|x\cdot \nabla \textbf{u}_{x_0}-\kappa \textbf{u}_{x_0}|^2\,dS_xd\sigma . \end{aligned}$$

Since $0<r<R<\infty$ are arbitrary, we conclude that $x\cdot \nabla \textbf{u}_{x_0}-\kappa \textbf{u}_{x_0}=0$, or $\textbf{u}_{x_0}$ is $\kappa$-homogeneous in ${{\mathbb {R}}}^n$. $\square$

Lemma 14

Let $\textbf{u}$ be an almost minimizer in $B_1$, $C_h$ a compact subset of ${\mathcal {R}}_\textbf{u}$, and $\delta$ as in Lemma 9. Then for every $x_0\in C_h$ there is a unique blowup $\textbf{u}_{x_0,0}\in {{\mathbb {H}}}_{x_0}$. Moreover, there exist $t_0>0$ and $C>0$ such that

$$\begin{aligned} \int _{\partial B_1}|\textbf{u}^\phi _{x_0,t}-\textbf{u}_{x_0,0}|\le Ct^{\delta /2} \end{aligned}$$

for all $0<t<t_0$ and $x_0\in C_h$.

Proof

By Lemmas 10 and 13,

$$\begin{aligned} \int _{\partial B_1}|\textbf{u}_{x_0,t}^\phi -\textbf{u}_{x_0,s}^\phi |\le Ct^{\delta /2},\quad s<t<t_0. \end{aligned}$$

By the definition of ${\mathcal {R}}_\textbf{u}$, for a subsequence of $t_j\rightarrow 0+$ we have $\textbf{u}_{x_0,t_j}\rightarrow \textbf{u}_{x_0,0}\in {{\mathbb {H}}}_{x_0}$. From $\lim _{t_j\rightarrow 0+}\frac{\phi (t)}{t^\kappa }=1$, we also have $\textbf{u}_{x_0,t_j}^\phi \rightarrow \textbf{u}_{x_0,0}$. Taking $s=t_j$ in the above inequality and passing to the limit, we get

$$\begin{aligned} \int _{\partial B_1}|\textbf{u}_{x_0,t}^\phi -\textbf{u}_{x_0,0}|\le Ct^{\delta /2},\quad t<t_0. \end{aligned}$$

To prove the uniqueness, we let $\tilde{\textbf{u}}_{x_0,0}$ be another blowup. Then

$$\begin{aligned} \int _{\partial B_1}|\tilde{\textbf{u}}_{x_0,0}-\textbf{u}_{x_0,0}|=0. \end{aligned}$$

We proved in Lemma 8 that every blowup is $\kappa$-homogeneous in ${{\mathbb {R}}}^n$, thus $\tilde{\textbf{u}}_{x_0,0}=\textbf{u}_{x_0,0}$ in ${{\mathbb {R}}}^n$. $\square$

As a consequence of the previous results, we prove the regularity of ${\mathcal {R}}_\textbf{u}$.

Proof of Theorem 3

The proof of the theorem is similar to those of Theorem 3 in [2] and Theorem 1.4 in [4].

Step 1. Let $x_0\in {\mathcal {R}}_\textbf{u}$. By Lemmas 12 and 14, there exists $\rho _0>0$ such that $B_{2\rho _0}(x_0)\subset B_1$, $B_{2\rho _0}(x_0)\cap \Gamma ^\kappa (\textbf{u})=B_{2\rho _0}(x_0)\cap {\mathcal {R}}_\textbf{u}$ and

$$\begin{aligned} \int _{\partial B_1}|\textbf{u}_{x_1,r}^\phi -\beta _{x_1}\max (x\cdot \nu (x_1),0)^\kappa \textbf{e}(x_1)|\le Cr^{\delta /2}, \end{aligned}$$

for any $x_1\in \Gamma ^\kappa (\textbf{u})\cap \overline{B_{\rho _0}(x_0)}$ and for any $0<r<\rho _0$. We then claim that $x_1\longmapsto \nu (x_1)$ and $x_1\longmapsto \textbf{e}(x_1)$ are Hölder continuous of order $\gamma$ on $\Gamma ^\kappa (\textbf{u})\cap \overline{B_{\rho _1}(x_0)}$ for some $\gamma =\gamma (n,\alpha ,q,\eta )>0$ and $\rho _1\in (0,\rho _0)$. Indeed, we observe that for $x_1$ and $x_2$ near $x_0$ and for small $r>0$,

$$\begin{aligned}&\int _{\partial B_1}|\beta _{x_1}\max (x\cdot \nu (x_1),0)^\kappa \textbf{e}(x_1)-\beta _{x_2}\max (x\cdot \nu (x_2),0)^\kappa \textbf{e}(x_2)|\,dS_x\\&\quad \le 2Cr^{\delta /2}+\int _{\partial B_1}|\textbf{u}_{x_1,r}^\phi -\textbf{u}_{x_2,r}^\phi |\\&\quad \le 2Cr^{\delta /2}+\frac{1}{\phi (r)}\int _{\partial B_1}\int _0^1|\nabla \textbf{u}(rx+(1-t)x_1+tx_2)||x_1-x_2|\,dt\,dS_x\\&\quad \le 2Cr^{\delta /2}+C\frac{|x_1-x_2|}{r^\kappa }. \end{aligned}$$

Moreover,

$$\begin{aligned}&\int _{\partial B_1}|\beta _{x_1}\max (x\cdot \nu (x_2),0)^\kappa \textbf{e}(x_2)-\beta _{x_2}\max (x\cdot \nu (x_2),0)^\kappa \textbf{e}(x_2)|\,dS_x\\&\quad \le |\beta _{x_1}-\beta _{x_2}|\int _{\partial B_1}\max (x\cdot \nu (x_2),0)^\kappa |\textbf{e}(x_2)|\\&\quad \le C|\lambda _+(x_1)^{\kappa /2}-\lambda _+(x_2)^{\kappa /2}|\le C|x_1-x_2|^\alpha . \end{aligned}$$

The above two estimates give

$$\begin{aligned}&\beta _{x_1}\int _{\partial B_1}|\max (x\cdot \nu (x_1),0)^\kappa \textbf{e}(x_1)-\max (x\cdot \nu (x_2),0)^\kappa \textbf{e}(x_2)|\,dS_x\\&\quad \le C\left( r^{\delta /2}+\frac{|x_1-x_2|}{r^\kappa }+|x_1-x_2|^\alpha \right) . \end{aligned}$$

Taking $r=|x_1-x_2|^{\frac{2}{\delta +2\kappa }}$, we get

$$\begin{aligned} \int _{\partial B_1}|\max (x\cdot \nu (x_1),0)^\kappa \textbf{e}(x_1)-\max (x\cdot \nu (x_2),0)^\kappa \textbf{e}(x_2)|\,dS_x\le C|x_1-x_2|^\gamma , \end{aligned}$$

where $\gamma =\min \{\frac{\delta }{\delta +2\kappa },\alpha \}$. Combining this with the following estimate (see equation (21) in the proof of Theorem 1.4 in [4])

$$\begin{aligned}&|\nu (x_1)-\nu (x_2)|+|\textbf{e}(x_1)-\textbf{e}(x_2)|\\&\quad \le C\int _{\partial B_1}|\max (x\cdot \nu (x_1),0)^\kappa \textbf{e}(x_1)-\max (x\cdot \nu (x_2),0)^\kappa \textbf{e}(x_2)|, \end{aligned}$$

we obtain the Hölder continuity of $x_1\longmapsto \nu (x_1)$ and $x_1\longmapsto \textbf{e}(x_1)$.

Step 2. We claim that for every $\varepsilon \in (0,1)$, there exists $\rho _\varepsilon \in (0,\rho _1)$ such that for $x_1\in \Gamma ^\kappa (\textbf{u})\cap \overline{B_{\rho _1}(x_0)}$ and $y\in \overline{B_{\rho _\varepsilon }(x_1)}$,

$$\begin{aligned}&\textbf{u}(y)=0\quad \text {if } (y-x_1)\cdot \nu (x_1)<-\varepsilon |y-x_1|, \end{aligned}$$

(7.6)

$$\begin{aligned}&|\textbf{u}(y)|>0\quad \text {if }(y-x_1)\cdot \nu (x_1)>\varepsilon |y-x_1|. \end{aligned}$$

(7.7)

Indeed, if (7.6) does not hold, then we can take a sequence $\Gamma ^\kappa (\textbf{u})\cap \overline{B_{\rho _1}(x_0)}\ni x_j\rightarrow \bar{x}$ and a sequence $y_j-x_j\rightarrow 0$ as $j\rightarrow \infty$ such that

$$\begin{aligned} |\textbf{u}(y_j)|>0\quad \text {and}\quad (y_j-x_j)\cdot \nu (x_j)<-\varepsilon |y_j-x_j|. \end{aligned}$$

Then we consider $\textbf{u}_j(x):=\frac{\textbf{u}(x_j+|y_j-x_j|x)}{|y_j-x_j|^\kappa }$ and observe that for $z_j:=\frac{y_j-x_j}{|y_j-x_j|}\in \partial B_1$, $|\textbf{u}_j(z_j)|>0$ and $z_j\cdot \nu (x_j)<-\varepsilon |z_j|$. As we have seen in the proof of Lemma 13, the rescalings $\textbf{u}_j$ at $x_j\in {\mathcal {R}}_\textbf{u}$ converge in $C^1_{{\text {loc}}}({{\mathbb {R}}}^n)$ to a $\kappa$-homogeneous solution $\textbf{u}_0$ of $\Delta \textbf{u}_0=f({\bar{x}},\textbf{u}_0)$. By applying Lemma 14 to $\textbf{u}_j$, we can see that $\textbf{u}_0(x)=\beta _{\bar{x}}\max (x\cdot \nu (\bar{x}),0)^\kappa \textbf{e}(\bar{x})$. Then, for $K:=\{z\in \partial B_1:\,z\cdot \nu (\bar{x})\le -\varepsilon /2|z|\}$, we have that $z_j\in K$ for large j by Step 1. We also consider a bigger compact set $\tilde{K}:=\{z\in {{\mathbb {R}}}^n\,:\,1/2\le |z|\le 2, z\cdot \nu (\bar{x})\le -\varepsilon /4|z|\}$, and let $t:=\min \{{\text {dist}}(K,\partial \tilde{K}),r_0\}$, where $r_0=r_0(n,\alpha ,q,M)$ is as in Lemma 6, so that $B_t(z_j)\subset \tilde{K}$. By applying Lemma 7, we obtain

$$\begin{aligned} \sup _{\tilde{K}}\left( |\nabla \textbf{u}_j|^2+|\textbf{u}_j|^{q+1}\right) \ge c(n,t)\int _{B_t(z_j)}\left( |\nabla \textbf{u}_j|^2+|\textbf{u}_j|^{q+1}\right) \ge c(n,\alpha ,q,M,\varepsilon ), \end{aligned}$$

which gives

$$\begin{aligned} \sup _{\tilde{K}}\left( |\nabla \textbf{u}_0|+|\textbf{u}_0|^{q+1}\right) >0. \end{aligned}$$

However, this is a contradiction since $\textbf{u}_0(x)=\beta _{\bar{x}}\textbf{e}(\bar{x})\max (x\cdot \nu (\bar{x}),0)^\kappa =0$ in $\tilde{K}$.

On the other hand, if (7.7) is not true, then we take a sequence $\Gamma (\textbf{u})\cap \overline{B_{\rho _1}(x_0)}\ni x_j\rightarrow \bar{x}$ and a sequence $y_j-x_j\rightarrow 0$ such that $|\textbf{u}(y_j)|=0$ and $(y_j-x_j)\cdot \nu (x_j)>\varepsilon |y_j-x_j|$. For $\textbf{u}_j$, $\textbf{u}_0(x)=\beta _{\bar{x}}\max (x\cdot \nu (\bar{x}),0)^\kappa \textbf{e}(\bar{x})$ and $z_j$ as above, we will have that $\textbf{u}_j(z_j)=0$ and $z_j\in K':=\{z\in \partial B_1\,:\,z\cdot \nu (\bar{x})\ge \varepsilon /2|z|\}$. Over a subsequence $z_j\rightarrow z_0\in K'$ and we have $\textbf{u}_0(z_0)=\lim _{j\rightarrow \infty }\textbf{u}_j(z_j)=0$. This is a contradiction since the half-space solution $\textbf{u}_0$ is nonzero in $K'$.

Step 3. By rotations we may assume that $\nu (x_0)=\textbf{e}_n$ and $\textbf{e}(x_0)=\textbf{e}_1$. Fixing $\varepsilon =\varepsilon _0$, by Step 2 and the standard arguments, we conclude that there exists a Lipschitz function $g:{{\mathbb {R}}}^{n-1}\rightarrow {{\mathbb {R}}}$ such that for some $\rho _{\varepsilon _0}>0$,

$$\begin{aligned}&B_{\rho _{\varepsilon _0}}(x_0)\cap \{\textbf{u}=0\}=B_{\rho _{\varepsilon _0}}(x_0)\cap \{x_n\le g(x')\},\\&B_{\rho _{\varepsilon _0}}(x_0)\cap \{|\textbf{u}|>0\}=B_{\rho _{\varepsilon _0}}(x_0)\cap \{x_n> g(x')\}. \end{aligned}$$

Now, taking $\varepsilon \rightarrow 0$, we can see that $\Gamma (\textbf{u})$ is differentiable at $x_0$ with normal $\nu (x_0)$. Recentering at any $y\in B_{\rho _{\varepsilon _0}}(x_0)\cap \Gamma (\textbf{u})$ and using the Hölder continuity of $y\longmapsto \nu (y)$, we conclude that g is $C^{1,\gamma }$. This completes the proof. $\square$

Appendix A: Example of almost minimizers

In this section we give an example of almost minimizers, following the line of Example 1 in [2]. We remark that a minimizer of (1.1) studied in [4] is a trivial example of almost minimizers.

Example 1

Let $\textbf{u}$ be a solution of the system

$$\begin{aligned} \Delta \textbf{u}+\textbf{b}(x)\cdot \nabla \textbf{u}-\lambda _+(x)|\textbf{u}^+|^{q-1}\textbf{u}^++\lambda _-(x)|\textbf{u}^-|^{q-1}\textbf{u}^-=0\quad \text {in }B_1, \end{aligned}$$

where $\textbf{b}\in L^p(B_1)$, $p>n$, is the velocity field. Then $\textbf{u}$ is an almost minimizer of the functional $\int \left( |\nabla \textbf{u}|^2+2F(x,\textbf{u})\right) \,dx$ with a gauge function $\omega (r)=C(n,p,\Vert \textbf{b}\Vert _{L^p(B_1)})r^{1-n/p}$.

Proof

Let $B_r(x_0)\Subset B_1$ and $\textbf{v}\in \textbf{u}+W^{1,2}_0(B_r(x_0))$. Without loss of generality assume that $x_0=0$. Then

$$\begin{aligned}&\int _{B_r}\nabla \textbf{u}\cdot \nabla (\textbf{u}-\textbf{v})\\&\quad =\int _{B_r}\textbf{b}\cdot \nabla \textbf{u}\cdot (\textbf{u}-\textbf{v})-\lambda _+|\textbf{u}^+|^{q-1}\textbf{u}^+\cdot (\textbf{u}-\textbf{v})+\lambda _-|\textbf{u}^-|^{q-1}\textbf{u}^-\cdot (\textbf{u}-\textbf{v})\\&\quad =\int _{B_r}\textbf{b}\cdot \nabla \textbf{u}\cdot (\textbf{u}-\textbf{v})-\lambda _+|\textbf{u}^+|^{q+1}-\lambda _-|\textbf{u}^-|^{q+1}\\ {}&\qquad +\lambda _+|\textbf{u}^+|^{q-1}\textbf{u}^+\cdot \textbf{v}-\lambda _-|\textbf{u}^-|^{q-1}\textbf{u}^-\cdot \textbf{v}. \end{aligned}$$

To estimate the integral in the last line, we note that the following estimate was obtained in the proof of Example 1 in [2]:

$$\begin{aligned} \int _{B_r}\textbf{b}\cdot \nabla \textbf{u}\cdot (\textbf{u}-\textbf{v})\le C(n,p,\Vert {\textbf {b}}\Vert _{L^p(B_1)})r^\gamma \int _{B_r}\left( |\nabla \textbf{u}|^2+|\nabla \textbf{v}|^2\right) ,\qquad \gamma =1-n/p. \end{aligned}$$

In addition, from $\textbf{u}^+\cdot \textbf{v}\le \textbf{u}^+\cdot \textbf{v}^+\le |\textbf{u}^+||\textbf{v}^+|$ and $-\textbf{u}^-\cdot \textbf{v}\le \textbf{u}^-\cdot \textbf{v}^-\le |\textbf{u}^-||\textbf{v}^-|$, we have

$$\begin{aligned}&\lambda _+|\textbf{u}^+|^{q-1}\textbf{u}^+\cdot \textbf{v}-\lambda _-|\textbf{u}^-|^{q-1}\textbf{u}^-\cdot \textbf{v}\\&\quad \le \lambda _+|\textbf{u}^+|^q|\textbf{v}^+|+\lambda _-|\textbf{u}^-|^q|\textbf{v}^-|\\&\quad \le \lambda _+\left( \frac{q}{1+q}|\textbf{u}^+|^{1+q}+\frac{1}{1+q}|\textbf{v}^+|^{1+q}\right) +\lambda _-\left( \frac{q}{1+q}|\textbf{u}^-|^{1+q}+\frac{1}{1+q}|\textbf{v}^-|^{1+q}\right) , \end{aligned}$$

and thus

$$\begin{aligned}&-\lambda _+|\textbf{u}^+|^{q+1}-\lambda _-|\textbf{u}^-|^{q+1}+\lambda _+|\textbf{u}^+|^{q-1}\textbf{u}^+\cdot \textbf{v}-\lambda _-|\textbf{u}^-|^{q-1}\textbf{u}^-\cdot \textbf{v}\\&\quad \le \frac{\lambda _+}{1+q}\left( |\textbf{v}^+|^{1+q}-|\textbf{u}^+|^{1+q}\right) +\frac{\lambda _-}{1+q}\left( |\textbf{v}^-|^{1+q}-|\textbf{u}^-|^{1+q}\right) . \end{aligned}$$

Combining the above equality and inequalities yields

$$\begin{aligned} \int _{B_r}\nabla \textbf{u}\cdot \nabla (\textbf{u}-\textbf{v})&\le Cr^\gamma \int _{B_r}\left( |\nabla \textbf{u}|^2+|\nabla \textbf{v}|^2\right) +\frac{1}{1+q}\int _{B_r}\lambda _+\left( |\textbf{v}^+|^{1+q}-|\textbf{u}^+|^{1+q}\right) \\&\quad +\frac{1}{1+q}\int _{B_r}\lambda _-\left( |\textbf{v}^-|^{1+q}-|\textbf{u}^-|^{1+q}\right) . \end{aligned}$$

From

$$\begin{aligned} \int _{B_r}\nabla \textbf{u}\cdot \nabla (\textbf{u}-\textbf{v})=\int _{B_r}|\nabla \textbf{u}|^2-\nabla \textbf{u}\cdot \nabla \textbf{v}\ge 1/2\int _{B_r}\left( |\nabla \textbf{u}|^2-|\nabla \textbf{v}|^2\right) , \end{aligned}$$

we further have

$$\begin{aligned}&\int _{B_r}(1/2-Cr^\gamma )|\nabla \textbf{u}|^2+\frac{1}{1+q}\left( \lambda _+|\textbf{u}^+|^{1+q}+\lambda _-|\textbf{u}^-|^{1+q}\right) \\&\quad \le \int _{B_r}(1/2+Cr^\gamma )|\nabla \textbf{v}|^2+\frac{1}{1+q}\left( \lambda _+|\textbf{v}^+|^{1+q}+\lambda _-|\textbf{v}^-|^{1+q}\right) . \end{aligned}$$

This implies that

$$\begin{aligned} (1-Cr^\gamma )\int _{B_r}\left( |\nabla \textbf{u}|^2+2F(x,\textbf{u})\right) \le (1+Cr^\gamma )\int _{B_r}\left( |\nabla \textbf{v}|^2+2F(x,\textbf{v})\right) , \end{aligned}$$

and hence we conclude that for $0<r<r_0$

$$\begin{aligned} \int _{B_r}\left( |\nabla \textbf{u}|^2+2F(x,\textbf{u})\right) \,dx\le (1+Cr^\gamma )\int _{B_r}\left( |\nabla \textbf{v}|^2+2F(x,\textbf{v})\right) \,dx, \end{aligned}$$

with $r_0$ and C depending only on n, p, $\Vert \textbf{b}\Vert _{L^p(B_1)}$. $\square$