Boundary regularity for viscosity solutions of nonlinear singular elliptic equations

Abstract

In this paper we establish boundary regularity for viscosity solutions of nonlinear singular elliptic equations

$$\begin{aligned} \left\{ \begin{array}{rclll} F\left( D^2u,Du,u,\frac{u_{x_n}}{x_n},\frac{u}{x_n^2},x\right) &{}=&{}f(x)&{}\textrm{in}&{}B'_1\times (0,1),\\ u(x)&{}=&{}\phi (x')&{}\textrm{on}&{}B'_1\times \{0\}. \end{array}\right. \end{aligned}$$

We prove that if there exists a smooth approximation solution $u_*$ which satisfies $u-u_*=O(x_n^\mu )$, then for any $\epsilon >0$, $u\in C^{\mu -\epsilon }$ up to the boundary $\{x_n=0\}$.

Appendices

Appendix A

We give the proof of Proposition 2.1, which is based on the following estimate of the oscillation of a flat solution [23, Proposition 3.3]. Recall the hypotheses $(\mathrm {H1^\circ })$–$(\mathrm {H3^\circ })$ in Sect. 2.

Proposition A.1

Assume $F\in C^1$ in $\mathcal {U}_{1,\delta }$, satisfies $(\mathrm {H1^\circ })$, $(\mathrm {H2^\circ })$, and

$$\begin{aligned} F(0,0,0,x)=0,\quad \quad |F_p(M,p,z,x)|\le \kappa \quad \text {in}\; \mathcal {U}_{1,\delta }. \end{aligned}$$

There exists small constants $c,\nu ,\beta$, depending only on $n,\lambda ,\Lambda$ and $\kappa$, such that if u is a viscosity solution of (2.1) with $u(0)=0$ and for some k

$$\begin{aligned} \Vert u\Vert _{L^\infty (B_1)}\le \delta '\le c2^{-2k}\delta ,\quad \quad |F(D^2u,Du,u,x)|\le \nu \delta ' \end{aligned}$$

and

$$\begin{aligned} |F_z(M,p,z,x)|\le \nu \quad \text {in} \,\,\,\, \mathcal {U}_{1,\delta }, \end{aligned}$$

then

$$\begin{aligned} \Vert u\Vert _{L^\infty (B_{\rho })}\le 2\rho ^\beta \delta ',\quad \text {for any}\; \rho \ge 2^{-k-1}. \end{aligned}$$

Proof of Proposition 2.1

Let u be a viscosity solution of (2.1) with f satisfying (2.2). In this proof we refer to positive constants depending only on $n,\lambda ,\Lambda ,\alpha _0,\alpha , K$ and $\psi$ as universal constants.

Let

$$\begin{aligned} w(x)=\frac{1}{r^{2+\alpha }}(u(rx)-P(N,q,t,rx)). \end{aligned}$$

The function $v:=w-w(0)$ is a solution of

$$\begin{aligned} {\tilde{F}}(D^2v,Dv,v,x)={\tilde{f}}(x),\quad \quad \Vert v\Vert _{L^\infty (B_1)}\le 2, \end{aligned}$$

where

$$\begin{aligned} {\tilde{F}}(M,p,z,x):= & {} r^{-\alpha }\left[F(r^\alpha M+N, r^{1+\alpha }p+q+rNx, r^{2+\alpha }(z+w(0))+P(N,q,t,rx),rx)\right.\\{} & {} -\left.F(N,q+rNx,r^{2+\alpha }w(0)+P(N,q,t,rx),rx)\right] \end{aligned}$$

and

$$\begin{aligned} {\tilde{f}}(x)=r^{-\alpha }\left[f(rx)-F(N,q+rNx,r^{2+\alpha }w(0)+P(N,q,t,rx),rx)\right]. \end{aligned}$$

By $(\mathrm {H3^\circ })$,

$$\begin{aligned} {|}{\tilde{f}}(x)|= & {} r^{-\alpha }\left| f(rx)-f(0)-\left[ F(N,q+rNx,r^{2+\alpha }w(0)\right. \right. \\{} & {} \left. \left. +P(N,q,t,rx),rx)-F(N,q,t,0)\right] \right| \\\le & {} [f]_{C^{0,\alpha _0}}r^{\alpha _0-\alpha }+K\left(|rNx|+|r^{2+\alpha }w(0)+P(N,q,t,rx)-t|+|rx|\right)\\\le & {} C(n)Kr^{\alpha _0-\alpha }. \end{aligned}$$

Obviously ${\tilde{F}}(0,0,0,x)=0$ and ${\tilde{F}}$ satisfies $(\mathrm {H1^\circ })$, $(\mathrm {H2^\circ })$ with ${\tilde{\delta }}:=r^{-\alpha }\delta /4$. By $(\mathrm {H3^\circ })$ we have

$$\begin{aligned} r|{\tilde{F}}_p(M,p,z,x)|+|{\tilde{F}}_z(M,p,z,x)|\le Kr^2\quad \text {for}\Vert M\Vert +|p|+|z|\le {\tilde{\delta }}. \end{aligned}$$

Choose $r\le r_0$, where $r_0$ is small depending only on $\delta$ and universal constants, and then $k\ge 1$ such that $2^{2k+1}\le cr^{-\alpha }\delta <2^{2k+3}$, where c is universal, given by Proposition A.1, then

$$\begin{aligned} \Vert v\Vert _{L^\infty (B_1)}\le 2\le c2^{-2k}{\tilde{\delta }}. \end{aligned}$$

By Proposition A.1,

$$\begin{aligned} \Vert v\Vert _{L^\infty (B_\rho )}\le 4\rho ^\beta \quad \text {for}\; \rho \ge 2^{-k-1} \end{aligned}$$

for some $\beta >0$ universal. Replacing v with $w-w(x_0)$ for $x_0\in B_{1/2}$, we conclude that

$$\begin{aligned} |w(x_1)-w(x_2)|\le 4|x_1-x_2|^\beta \quad \text {for any}\; x_1,x_2\in B_{1/2}, |x_1-x_2|\ge C\delta ^{-\frac{1}{2}}r^{\frac{\alpha }{2}} \end{aligned}$$

(A.1)

for some $C>0$ universal.

We next prove that there exist $\eta \in (0,1)$ universal and ${\tilde{N}},{\tilde{q}},{\tilde{t}}$ such that

$$\begin{aligned} \Vert w-P({\tilde{N}},{\tilde{q}},{\tilde{t}},x)\Vert _{L^\infty (B_\eta )}\le \eta ^{2+\alpha } \end{aligned}$$

(A.2)

with

$$\begin{aligned} F(r^\alpha {\tilde{N}}+N,r^{1+\alpha }{\tilde{q}}+q,r^{2+\alpha }{\tilde{t}}+t,0)=f(0). \end{aligned}$$

(A.3)

This implies the conclusion of Proposition 2.1.

Assume by contradiction that the claim does not hold. Then there exists a sequence of $r_k\rightarrow 0$ and corresponding $F_k,f_k,w_k,N_k,q_k,t_k$ for which no ${\tilde{N}},{\tilde{q}},{\tilde{t}}$ satisfy (A.2) and (A.3). By $({\textrm{H3}}^\circ )$, (2.2) and (A.1), there exists a subsequence $\{k_j\}_{j\ge 1}$, which we still denote by $\{k\}_{k\ge 1}$, such that

$$\begin{aligned} F_k\rightarrow F_*,\quad D_MF_k\rightarrow D_MF_*,\quad f_k\rightarrow f_*\quad \textrm{uniformly}, \\ w_k\rightarrow w_*\quad \text {uniformly in}B_{1/2} \\ N_k\rightarrow N_*,\quad q_k\rightarrow q_*,\quad t_k\rightarrow t_*. \end{aligned}$$

We claim that $w_*$ is viscosity solution of

$$\begin{aligned} \sum _{i,j}\frac{\partial F_*}{\partial M_{ij}}(N_*,q_*,t_*,0)D_{ij}w_*=0. \end{aligned}$$

(A.4)

Assume by contradiction that there is a smooth function $\varphi +\epsilon |x-x_*|^2$ which touches $w_*$ from below at $x_*\in B_{1/2}$ and

$$\begin{aligned} \sum _{i,j}\frac{\partial F_*}{\partial M_{ij}}(N_*,q_*,t_*,0)D_{ij}\varphi (x_*)>\epsilon >0. \end{aligned}$$

Then $\varphi +const$ touches $w_k$ from below at $x_k$ with $x_k\rightarrow x_*$. Hence,

$$\begin{aligned} Kr_k^{\alpha _0-\alpha }\ge & {} r_k^{-\alpha }(f_k(r_kx_k)-f_k(0))\\\ge & {} \left[ F_k\!\left( r_k^\alpha D^2\!\varphi (x_k)+N_k, r_k^{1+\alpha }D\varphi (x_k)+q_k+r_kN_kx_k, r_k^{2+\alpha }(\varphi (x_k)+c)\right. \right. \\{} & {} \left. \left. +P(N_k,q_k,t_k,r_kx_k),r_kx_k\right) -F_k(N_k,q_k,t_k,0)\right] \cdot r_k^{-\alpha }\\\ge & {} \sum _{i,j}\frac{\partial F_k}{\partial M_{ij}}(N_k,q_k,t_k,0)D_{ij}\varphi (x_k)-C(\varphi )\psi \left(C(\varphi )r_k^\alpha \right)\\{} & {} -C(K,\varphi )r_k^{1-\alpha }>\epsilon /2\quad \text {for}\;k\;\text {large}, \end{aligned}$$

which is a contradiction as $k\rightarrow \infty$. Hence (A.4) is true, which implies that there exist $N',q',t'$ and $\eta \in (0,1)$ universal such that

$$\begin{aligned} \sum _{i,j}\frac{\partial F_*}{\partial M_{ij}}(N_*,q_*,t_*,0)N'_{ij}=0 \end{aligned}$$

(A.5)

and

$$\begin{aligned} \Vert w_*-P(N',q',t',x)\Vert _{L^\infty (B_\eta )}\le \frac{\eta ^{2+\alpha }}{3}. \end{aligned}$$

(A.6)

We claim that there exists $s_k\in {\mathbb {R}}$ such that for ${\tilde{N}}_k=N'+s_kI_n$ we have

$$\begin{aligned} I_k:=F_k(r_k^\alpha {\tilde{N}}_k+N_k,r_k^{1+\alpha }q'+q_k,r_k^{2+\alpha }t'+t_k,0)-f_k(0)=0. \end{aligned}$$

Indeed,

$$\begin{aligned} r_k^{-\alpha }I_k= & {} r_k^{-\alpha }\left[ F_k(r_k^\alpha {\tilde{N}}_k+N_k,r_k^{1+\alpha }q'+q_k,r_k^{2+\alpha }t'+t_k,0)\right. \\{} & {} \quad \left. -F_k(r_k^\alpha N'+N_k,r_k^{1+\alpha }q'+q_k,r_k^{2+\alpha }t'+t_k,0)\right. \\{} & {} \quad +\left. F_k(r_k^\alpha N'+N_k,r_k^{1+\alpha }q'+q_k,r_k^{2+\alpha }t'+t_k,0)-F_k(N_k,q_k,t_k,0)\right] \\= & {} J_k+\sum _{i,j}\frac{\partial F_k}{\partial M_{ij}}(N_k,q_k,t_k,0)N'_{ij}+O\left(\psi \left(Cr_k^\alpha \right)\right)+O(r_k), \end{aligned}$$

where $C>0$ depends only on $n,\lambda$ and $\Lambda$, and

$$\begin{aligned} \Lambda |s_k|\ge |J_k|\ge \lambda |s_k|. \end{aligned}$$

Hence by (A.5), for k large there exists $|s_k|\le \frac{\eta ^{2+\alpha }}{3}$ such that $r_k^{-\alpha }I_k=0$. It follows that

$$\begin{aligned}{} & {} \Vert w_k-P({\tilde{N}}_k,q',t',x)\Vert _{L^\infty (B_\eta )}\le \Vert w_k-w_*\Vert _{L^\infty (B_{1/2})} +\Vert w_*\\{} & {} \quad -P(N',q',t',x)\Vert _{L^\infty (B_\eta )}+|s_k|\le \eta ^{2+\alpha } \end{aligned}$$

by (A.6). We reach a contradiction. $\square$

Appendix B

In this section give the proof of Theorems 4.1 and 4.3 in Sect. 4. Recall the hypotheses $\mathrm {(H4)}$ and $\mathrm {(H5)}$ on F in these theorems. Also, in Theorem 4.1 we assume $F\in \mathcal {A}^1$ (i.e. F satisfies $\mathrm {(H1)}$–$\mathrm {(H3)}$ in $\mathcal {U}_\delta =\mathcal {U}_\delta (0)$ with $m=1$); in Theorem 4.3 we assume $F\in \mathcal {A}^m$ where $\mu \in (m+1,m+2]$. We refer to positive constants depending only on the constants $n,\lambda ,\Lambda ,K,b$ and $\mu$ in the assumptions of these theorems as universal constants.

Using the hypotheses $\mathrm {(H5)}$, and $\mathrm {(H2)}$, $\mathrm {(H3)}$ in $\mathcal {U}_\delta$, it is straightforward to prove the lemma below. Recall the notation ${\bar{p}}=(p,z)$ in (4.1).

Lemma B.1

Assume $f\in C^{0,1}({\bar{\Omega }})$ and

$$\begin{aligned} |f(x',0)|\le b\delta /8\quad \text {for}\; x'\in B'_1. \end{aligned}$$

Then for each $(a,{\bar{p}},x')\in U_{\!c_0\delta }:=\{\Vert a\Vert ,|{\bar{p}}|\le c_0\delta , x'\!\!\in \!B'_1\}$, the equation

$$\begin{aligned} g(a,r,{\bar{p}},x')=f(x',0) \end{aligned}$$

has a unique solution $r=\xi (a,{\bar{p}},x')$ with

$$\begin{aligned} |\xi (a,{\bar{p}},x')|\le C\left(\Vert a\Vert +|{\bar{p}}|+|f(x',0)|\right), \end{aligned}$$

(B.1)

where $c_0\in (0,1), C>0$ are universal. Moreover, the function $r=\xi (a,{\bar{p}},x')$ is increasing and Lipschitz continuous with respect to a; Furthermore, for any $(a_i,{\bar{p}}_i,x_i)\in U_{\!c_0\delta }, i=1,2$ with $a_1<a_2$ (i.e., $a_2-a_1$ is a positive, symmetric matrix), we have

$$\begin{aligned} 0<\xi (a_2,{\bar{p}}_2,x'_2)-\xi (a_1,{\bar{p}}_1,x'_1)\le C\Vert a_2-a_1\Vert \end{aligned}$$

(B.2)

provided that

$$\begin{aligned} C\left\rbrace |{\bar{p}}_2-{\bar{p}}_1|+(\Vert a_1\Vert +|{\bar{p}}_1|+\Vert f\Vert _{C^{0,1}})|x'_2-x'_1|\right\lbrace \le \Vert a_2-a_1\Vert , \end{aligned}$$

where $C>0$ is universal.

Proof of Theorem 4.2

Using Lemma B.1 and a barrier argument as in [19, p. 595], we can prove that

$$\begin{aligned} |u(x)-\phi (x')|\le C\sigma ^{1/3}t^2,\quad x'\in B'_{1/2}, t\in (0,1) \end{aligned}$$

for some $C>0$ universal. This combined with Theorem 3.1 implies the $C^{1,1}$ regularity for small viscosity solutions of (1.1) up to the boundary, i.e., Theorem 4.2.

Proof of Theorem 4.1

By Theorem 4.2, we can assume

$$\begin{aligned} \Vert u\Vert _{C^{1,1}({\bar{\Omega }})}+\left\Vert\frac{u_t}{t}\right\Vert_{L^\infty (\Omega )}+\Vert f\Vert _{C^{0,1}({\bar{\Omega }})} +\Vert \phi \Vert _{C^{2,\alpha }(B'_1)}\le \delta '\le c\delta , \end{aligned}$$

(B.3)

where $c>0$ is small, universal, to be chosen below. We can now reduce the Eq. (1.1) to

$$\begin{aligned} F_1\left(D^2u,\frac{u_t}{t},\frac{u}{t^2},x\right)=f \end{aligned}$$

where $F_1: \bar{\mathcal {D}}_1:=\mathcal {S}^n\times {\mathbb {R}}\times {\mathbb {R}}\times {\bar{\Omega }}\rightarrow {\mathbb {R}}$ is defined as

$$\begin{aligned} F_1(M,r,s,x):=F(M,Du(x),u(x),r,s,x). \end{aligned}$$

By (B.3), the operator $F_1$ satisfies $\mathrm {(H2)}$ with $\mathcal {U}_\delta$ replaced by

$$\begin{aligned} \mathcal {U}_{1,\delta /2}:=\{(M,r,s,x)\in \bar{\mathcal {D}}_1: \Vert M\Vert +|r|+|s|\le \delta /2\}. \end{aligned}$$

Since $F(\mathbf{{0}},x)=0$ and $F\in C^{1,1}(\mathcal {U}_\delta )$, by (B.3) we have

$$\begin{aligned} |F_1(M,r,s,x)-F_1(M,r,s,{\tilde{x}})|\le 4K\delta '|x-{\tilde{x}}|\quad \quad \text {for any}\; \Vert M\Vert +|r|+|s|\le \delta '.\nonumber \\ \end{aligned}$$

(B.4)

Denote

$$\begin{aligned} {\bar{p}}(x'):={\bar{p}}(x',0)=\left((D\phi (x'),0),\phi (x')\right). \ \end{aligned}$$

(B.5)

The operator $F_1$ also satisfies the monotonicity assumption for small a, r, i.e., the function $g_1$ defined by

$$\begin{aligned} g_1(a,r,x'):=F_1\left(\left[\begin{array}{cc} a&{}0\\ 0&{}r \end{array}\right],r,\frac{r}{2},x',0\right) \end{aligned}$$

satisfies $g_1(a,r,x')=g(a,r,{\bar{p}}(x'),x')$, where g is as in $\mathrm {(H5)}$. Hence,

$$\begin{aligned} \frac{\partial g_1(a,r,x')}{\partial r}\le -b<0\quad \quad \textrm{for}\quad \Vert a\Vert +|r|\le \delta /2, x'\in B'_1, \end{aligned}$$

Let $\xi$ be the function given by Lemma B.1 and $\xi (x'), \omega (x')$ be defined in (4.3) and (4.4) respectively. By Lemma B.1 and (B.3), we have

$$\begin{aligned} |\xi (x')|\le C\delta ' \end{aligned}$$

(B.6)

for some $C>0$ universal, hence $(\omega (x'),x',0)\in \mathcal {U}_\delta$ if $\delta '\le c\delta$ for some c small, universal, and $\omega (x')$ solves the equation

$$\begin{aligned} F(\omega (x'),x',0)=f(x',0). \end{aligned}$$

(B.7)

Using (B.7), Lemma B.1 and arguing as in [19, Theorem 3.7, Corollary 3.13], we obtain that

$$\begin{aligned} \left|u(x)-P_{x'_0}(x)\right|\le C\delta '\left(|x'-x'_0|+t\right)^{2+\alpha }\quad \text {for any}\; x'_0\in B'_{1/2}\;\textrm{and}\;x\in \Omega , \end{aligned}$$

(B.8)

where $\alpha \in (0,1), C>0$ are universal, and we choose $\delta '\le c\delta$ for some c small, universal, and denote

$$\begin{aligned} P_{x'_0}(x):=\phi (x'_0)+D\phi (x'_0)\cdot (x'-x'_0)+\frac{1}{2}(x'-x'_0)^TD^2\phi (x'_0)(x'-x'_0)+\frac{1}{2}\xi (x'_0)t^2\nonumber \\ \end{aligned}$$

(B.9)

Hence $u\in C^{2,\alpha }$ at each point on $B'_{1/2}\times \{0\}$. By (B.6) and the definition (1.2) we have

$$\begin{aligned} |\omega [P_0](x)|\le C\delta '\quad \text {for}x\in \Omega , \end{aligned}$$

where $P_0$ is defined in (B.9) with $x'_0=0$. By (B.7) we have $F\left(\omega [P_0](0),0\right)=f(0)$. It follows that

$$\begin{aligned}{} & {} |(F[P_0]-f)(x)|\\{} & {} \quad \le \left|F\left(\omega [P_0](x),x\right)-F\left(\omega [P_0](x),0\right)\right|+\left| F\left(\omega [P_0](x),0\right)\right. \\{} & {} \qquad \left. -F\left(\omega [P_0](0),0\right)\right| +|f(x)-f(0)|\\{} & {} \quad \le C\delta '|x|, \end{aligned}$$

for some C universal, where we use that $F(\mathbf{{0}},x)=0$. We have similar estimates for $P_{\!x'_0}$ with $x'_0\in B'_{1/2}$.

Choosing $\delta '\le \delta _0$, where $\delta _0$ is small (depending only on $\delta$ and universal constants), and applying Theorem 1 with $m=1$ (see Remark 3.1), we obtain that $u\in C^{2,\alpha }({\bar{\Omega }}_{1/2})$, and

$$\begin{aligned}{}[D^2u]_{C^{\alpha }({\bar{\Omega }}_{1/2})}\le \delta ^{'\beta }, \end{aligned}$$

where $\beta >0$ depends only on $\alpha$. $\square$

Proof of Theorem 4.3

By Theorem 4.2, we can assume

$$\begin{aligned} \Vert u\Vert _{C^{1,1}({\bar{\Omega }})}+\Vert \phi \Vert _{C^{m+1,1}(B'_1)}+\Vert f\Vert _{C^{m}({\bar{\Omega }})}\le \delta '\le c\delta \end{aligned}$$

(B.10)

for some $c>0$ small, universal, to be chosen below. Taking $x'=x'_0$ in (B.8) we obtain

$$\begin{aligned} \left|u(x)-\phi (x')-\frac{1}{2}\xi (x')t^2\right|\le C\delta 't^{2+\alpha }\quad \text {for any}x\in B'_{1/2}\times [0,1). \end{aligned}$$

(B.11)

This [or the polynomial $P_{0}$ in (B.9)] is a second order approximation of u at $x=0$. Using the induction argument in [9, Sect. 2] we can obtain a higher order approximation of u, i.e., there exists a polynomial $Q_0(x'\!,t)$ of degree $m+1$, which is of the form

$$\begin{aligned} Q_0(x'\!,t)=p_0(x')+t^2q_0(x'\!,t), \end{aligned}$$

where $p_0(x')$ is the Taylor expansion of $\phi$ at 0 of order $m+1$, $q_0(x'\!,t)$ is a polynomial of degree $m-1$ whose coefficients are bounded by $C_0\delta '$, satisfying

$$\begin{aligned} \left|D^j\left(F[Q_0]-f\right)(x)\right|\le C_0\delta '|x|^{m-j},\quad j=0,1,\ldots ,m, \end{aligned}$$

and

$$\begin{aligned} |(u-Q_0)(x)|\le C_0\delta '|x|^{2+\alpha },\quad \quad |(u-Q_0)(x',0)|\le C_0\delta '|x'|^{m+2}, \end{aligned}$$

where $C_0>0$ is universal.

For any $\epsilon >0$ small, set $\mu ^-:=\mu -\epsilon$. For $x\in E:=\{|x'|<\theta ^{\frac{1}{1+\epsilon _0}},0<t<\theta \}$, let

$$\begin{aligned} w(x)=A\delta '\left(|x'|^{2+2\epsilon _0}+t^2\right)^{\frac{\mu ^-}{2}},\quad \quad A:=8C_0\theta ^{2+\alpha /2-\mu ^-}, \end{aligned}$$

where $\theta >0$ is small, depending only on $\epsilon$ and universal constants, and we choose $\epsilon _0$ satisfying

$$\begin{aligned} 0<\epsilon _0<\frac{\epsilon }{\mu -\epsilon },\quad \quad \frac{2+\alpha }{1+\epsilon _0}>2+\alpha /2. \end{aligned}$$

(B.12)

This gives $w\ge |u-Q_0|$ on $\partial E$. Arguing as in [9, p. 122 (iv)] we obtain

$$\begin{aligned} |u-Q_0|\le w\le C\delta '|x|^{\mu ^-}\quad \quad \textrm{in}\quad {\bar{\Omega }}_{1/2}, \end{aligned}$$

where $C>0$ depends only on $\epsilon$ and universal constants. Replacing 0 with any $x'_0\in B'_{\rho _0}$, where $\rho _0>0$ is small depending only on $\mu ,\epsilon$ and universal constants, and using Theorem 1 (see Remark 3.1), we obtain Theorem 4.3.