Regularity theory for non-autonomous problems with a priori assumptions

Abstract

We study weak solutions and minimizers u of the non-autonomous problems ${\text {div}} A(x, Du)=0$ and $\min _v \int _\Omega F(x,Dv)\,dx$ with quasi-isotropic (p, q)-growth. We consider the case that u is bounded, Hölder continuous or lies in a Lebesgue space and establish a sharp connection between assumptions on A or F and the corresponding norm of u. We prove a Sobolev–Poincaré inequality, higher integrability and the Hölder continuity of u and Du. Our proofs are optimized and streamlined versions of earlier research that can more readily be further extended to other settings. Connections between assumptions on A or F and assumptions on u are known for the double phase energy $F(x, \xi )=|\xi |^p + a(x)|\xi |^q$. We obtain slightly better results even in this special case. Furthermore, we also cover perturbed variable exponent, Orlicz variable exponent, degenerate double phase, Orlicz double phase, triple phase, double variable exponent as well as variable exponent double phase energies and the results are new in most of these special cases.

1 Introduction

We consider the divergence form, quasilinear elliptic equation

and the corresponding F-energy minimization

where A and F have quasi-isotropic (p, q)-growth (see Definition 4.1). Since we allow A and F to depend on x, these are non-autonomous problems. The strategy for dealing with non-autonomous problems is often the reduction to and approximation with autonomous problems, such as the p-power energy $F(\xi ):=|\xi |^p$, $p\in (1,\infty )$, and the p-Laplace equation with $A(\xi ):=|\xi |^{p-2}\xi$. The maximal regularity of weak solutions already to the p-Laplace equation when $p\ne 2$ is $C^{1,\alpha }$ for some $\alpha \in (0,1)$ (e.g., [23, 29, 43, 46, 52]) and this is the objective also in more general cases, including in this article. The approximation technique is often used to deal with Marcellini’s [47] (p, q)-growth energies, $|\xi |^p \lesssim F(\xi )\lesssim |\xi |^q+1$ and $1<p\leqslant q$, provided that $\frac{q}{p}$ is close to 1, see, e.g., [9, 10, 21, 22, 48].

To explain the objective of the current paper we consider the double phase functional $F(x,\xi ):= |\xi |^p + a(x) |\xi |^q$ with $1<p\leqslant q$ and $a:\Omega \rightarrow [0, L_\omega ]$, which is a special case of (p, q)-growth. This model was first studied by Zhikov [53, 54] in the 1980’s and has recently enjoyed a resurgence after a series of papers by Baroni, Colombo and Mingione [6,7,8, 15,16,17]. They studied the relationship between the parameters p and q and the Hölder-exponent $\alpha$ of a and established maximal regularity of the minimizer u in the following three cases of a priori information:

• (ap1) $u\in W^{1,p}(\Omega )$ and $q-p\leqslant \frac{p}{n} \alpha$

• (ap2) $u\in L^\infty (\Omega )$ and $q-p\leqslant \alpha$

• (ap3) $u\in C^{0,\gamma }(\Omega )$ and $q-p< \frac{1}{1-\gamma } \alpha$

Furthermore, in the first two cases the inequality is sharp in the sense that there exist counter-examples to regularity which fail the inequality arbitrarily little [5, 28]. The case of equality in (ap3) is an open problem. Ok [50] added to these a fourth, likewise sharp, case:

• (ap4) $u\in L^{s^*}(\Omega )$ and $q-p\leqslant \frac{s}{n} \alpha$

In view of the Sobolev embedding when $s<n$, $s=n$ and $s>n$, this suggests the unifying, albeit slightly stronger, assumption $u\in W^{1,s}(\Omega )$ and $q-p\leqslant \frac{s}{n} \alpha$.

While the relationship between a priori information on u and the conditions for double phase F are quite well understood, this is not the case for the wide range of recently introduced double phase variants, which extend it or combine it with the variable exponent case $F(x,\xi )=|\xi |^{p(x)}$ [24, 51]. These variants include perturbed variable exponent, Orlicz variable exponent, degenerate double phase, Orlicz double phase, triple phase, double variable exponent and variable exponent double phase. See Corollary 1.1 for the corresponding expressions F and Table 1 for examples of our assumptions in some of these cases. We refer to [41] for references up to 2020 and [3, 4, 18, 20, 30, 32, 45, 49] for some more recent advances on variants of the variable exponent and double phase models.

In most of the special cases, both lower and maximal regularity remain unstudied under assumptions (ap2)–(ap4). Recently, Baasandorj and Byun [2] proved maximal regularity of the Orlicz triple phase case in a massive paper. Rather than study each case individually, we introduced an approach based on generalized Orlicz spaces in [41] and proved maximal regularity for minimizers when $F(x,\xi )=F(x,|\xi |)$ has so-called Uhlenbeck structure. In [42] we extended the results to weak solutions and minimizers of problems with (p, q)-growth without the Uhlenbeck restriction. In both articles we only considered the assumption $u\in W^{1,\varphi }(\Omega )$ corresponding to case (ap1). In this article we cover all the different assumptions from cases (ap1)–(ap4), including as special cases all the double phase variants listed in the previous paragraph.

We build on the harmonic approximation approach from [8]. Our method is more streamlined and we are even able to improve the results in the double phase case slightly by introducing the following version of (ap2), which is natural to expect based on the intuition of the Sobolev embedding, and a version of (ap3) with equality provided we have vanishing Hölder continuity:

• (ap2$'$) $u\in BMO(\Omega )$ and $q-p\leqslant \alpha$

• (ap3$'$) $u\in VC^{0,\gamma }(\Omega )$ and $q-p\leqslant \frac{1}{1-\gamma } \alpha$

This article represents a substantial generalization and unification of prior theory. We expect that our optimized methods can more readily be further extended to other settings.

In recent years, many papers consider bounded weak solutions or minimizers (i.e. Case (ap2)), see for instance [2, 12, 16, 34]. The boundedness can be naturally obtained from the maximum principle for bounded Dirichlet boundary value problems, and is thus a fundamental assumption. The following special case of Corollary 5.14 showcases our results for $L^\infty$. We emphasize that even many of these special case results are new and that our main results, Theorems 5.3 and 5.11 and Corollary 5.14, also cover other a priori assumptions and structures.

Corollary 1.1

(Bounded minimizers in special cases) Let $1<p< \min \{q,r\}$, the variable exponents p(x) and q(x) with $p(x)\leqslant q(x)$ be Hölder continuous and bounded away from 1 and $\infty$, and $a\in C^{0,\alpha _a}$ and $b\in C^{0,\alpha _b}$ be non-negative and bounded. Assume that $F(x,\xi )=f(x,|\xi |)$ equals one of following functions with corresponding additional conditions hold:

Model	f(x, t)	Additional condition
Perturbed double phase	$t^p + a(x) t^q\log (e+t)$	$\alpha _a\geqslant q-p$
Triple phase	$t^p + a(x) t^q + b(x) t^r$	$\alpha _a\geqslant q-p$ & $\alpha _b\geqslant r-p$
Variable exponent double phase	$t^{p(x)} + a(x) t^{q(x)}$	$\alpha _a(x)\geqslant q(x)-p(x)$

Then every minimizer $u\in W^{1,1}_{\textrm{loc}}(\Omega )\cap L^\infty (\Omega )$ of ($\min F$) satisfies $u\in C^{1,\alpha }_{\textrm{loc}}(\Omega )$ for some $\alpha \in (0,1)$ independent of $\Vert u\Vert _{L^\infty (\Omega )}$.

Remark 1.2

The previous corollary holds also with the weaker, but more difficult to check, assumption $u\in W^{1,1}_{\textrm{loc}}(\Omega )\cap BMO(\Omega )$. Without the additional a priori information $u\in BMO(\Omega )$, the additional conditions are

$$\begin{aligned} \alpha _a\geqslant \frac{n}{p}(q-p),\ \alpha _b\geqslant \frac{n}{p}(r-p), \text { and } \alpha _a(x)\geqslant \frac{n}{p(x)}(q(x)-p(x)). \end{aligned}$$

These are stronger assumptions when $p<n$ as expected, since if $p\geqslant n$, then $W^{1,p}_\textrm{loc}(\Omega ) \subset BMO_\textrm{loc}(\Omega )$, so the a priori assumption $u\in BMO(\Omega )$ actually contains no additional information.

Remark 1.3

Our results also apply in the following cases with the same assumptions as in Corollary 1.1.

Model	f(x, t)
Perturbed variable exponent	$t^{p(x)} \log (e+t)$
Orlicz variable exponent	$\varphi (t)^{p(x)}$ or $\varphi (t^{p(x)})$
Double variable exponent	$t^{p(x)}+t^{q(x)}$
Degenerate double phase	$t^p + a(x) t^p \log (e+t)$

However, in these cases the a priori information does not give any improvement in the result. The reason is that for these energies, a calculation shows that holds if and only if the condition holds, see Sect. 3. In other words, for these cases we obtain a new proof of results previously obtained in [42].

On the other hand, a priori information does matter for the Orlicz double phase $\varphi (t) + a(x) \psi (t)$, where $a\in C^{0,\lambda }$ and $\psi /\varphi$ is almost increasing, but the conditions get a bit messy. The main condition is that for each $\varepsilon >0$ there exists $\beta >0$ such that

$$\begin{aligned} \lambda (r) \frac{\psi (r^{-1/(1+\varepsilon )})}{\varphi (r^{-{1/(1+\varepsilon )}})} \lesssim r^\beta . \end{aligned}$$

The detailed calculations are left to the interested reader, cf. [41, Corollary 8.4].

We study regularity of weak solutions or minimizers u with the additional information that they belong to $L^{s^*}$, BMO, $L^\infty$ or $C^{0,\gamma }$, and study sharp conditions on A or F corresponding to restrictions on u. See Definition 3.1 for these sharp conditions and Example 3.2 for their interpretation in the double phase case. The functions $A:\Omega \times \mathbb {R} ^n\rightarrow \mathbb {R} ^n$ from (${\text {div}}A$) and $F:\Omega \times \mathbb {R} ^n \rightarrow \mathbb {R}$ from ($\min F$) have quasi-isotropic (p, q)-growth structure, given in Definition 4.1. We briefly explain the strategy and structure of the paper.

In Sect. 3, we consider lower order regularity in cases of generalized Orlicz growth and a priori information. We prove $C^{0,\alpha }$-regularity for some $\alpha \in (0,1)$ and higher integrability for quasiminimizers (Theorems 3.12 and 3.14). These are based on Sobolev–Poincaré type inequalities with a priori information, which are obtained in Theorem 3.4 with Lemma 3.8.

In Sect. 5 we prove the main results, maximal regularity of weak solutions and minimizers for cases (ap2)–(ap4). We prove $C^{0,\alpha }$-regularity for every $\alpha \in (0,1)$ and $C^{1,\alpha }$-regularity for some $\alpha \in (0,1)$ assuming a priori $C^{0,\gamma }$-information (Theorems 5.3 and 5.11). Other cases follow as corollaries by the lower order regularity results. The crucial step of the proofs is approximating the original problem (${\text {div}}A$) and ($\min F$) with a suitable autonomous problem and obtaining a comparison estimate between solutions to the original problem and the autonomous problem. For the approximation we use tools from [42], but the comparison is achieved quite differently from our earlier papers. In this paper, we use harmonic approximation in Lemma 4.13 generalizing the double phase case from [8]. The main innovations are inventing assumptions and formulating results optimally to cover all special cases while also being sharp, see comments before Theorem 5.3 for details.

We start in Sect. 2 by recalling notation, definitions and basic results on generalized Orlicz spaces.

2 Preliminaries and notation

Throughout the paper we always assume that $\Omega$ is a bounded domain in ${\mathbb {R}^n}$ with $n \geqslant 2$. For $x_0\in {\mathbb {R}^n}$ and $r>0$, $B_r(x_0)$ is the open ball with center $x_0$ and radius r. If its center is clear or irrelevant, we write $B_r=B_r(x_0)$. The characteristic function $\chi _E$ of $E\subset \mathbb {R} ^n$ is defined as $\chi _E(x)=1$ if $x\in E$ and $\chi _E(x)=0$ if $x\not \in E$.

Let $f,g:E \rightarrow \mathbb {R}$ be measurable in $E\subset {\mathbb {R}^n}$. We denote the integral average of f over E with $0<|E|<\infty$ by . The gradient of f is denoted Df. If $E\subset \mathbb {R}$, then f is said to be almost increasing with constant $L\geqslant 1$ if $f(s)\leqslant L f(t)$ whenever $s\leqslant t$. If $L=1$, f is increasing. Similarly, we define an almost decreasing or decreasing function. We write $f\lesssim g$, $f\approx g$ and $f\simeq g$ if there exists $C\geqslant 1$ such that $f(y)\leqslant Cg(y)$, $C^{-1}f(y)\leqslant g(y) \leqslant Cf(y)$ and $f(C^{-1}y)\leqslant g(y) \leqslant f(Cy)$ for all $y\in E$, respectively. We use c as a generic constant whose value may change between appearances.

A modulus of continuity $\omega :[0,\infty )\rightarrow [0,\infty )$ is concave and increasing with $\omega (0)=\lim _{r\rightarrow 0^+}\omega (r)=0$. We define the Hölder seminorm by

$$\begin{aligned} {[}u]_{\gamma }=[u]_{\gamma ,\Omega }:= \sup _{x,y\in \Omega , \, x\ne y } \frac{|u(x)-u(y)|}{|x-y|^\gamma } \qquad \text {and}\qquad [u]_{\gamma , r}:=\sup _{x\in \Omega } [u]_{\gamma , B_r(x)\cap \Omega }. \end{aligned}$$

Vanishing Hölder continuity $VC^{0,\gamma }(\Omega )$ means that $u\in C^{0,\gamma }(\Omega )$ and $\lim _{r\rightarrow 0^+}[u]_{\gamma ,r}=0$. The spaces $C^{0,\gamma (\cdot )}(\Omega )$ and $C^{0,\log }(\Omega )$, as well as their vanishing versions, are defined similarly with $|x-y|^{\gamma (x)}$ and $\log (e+\frac{1}{|x-y|})$ instead of $|x-y|^\gamma$ in the denominator.

We refer to [33, Chapter 2] for the following definitions and properties.

Definition 2.1

We define some conditions for $\varphi :\Omega \times [0,\infty ]\rightarrow [0,\infty )$ and $\gamma \in \mathbb {R}$ related to regularity with respect to the second variable, which are supposed to hold for all $x\in \Omega$ and a constant $L\geqslant 1$ independent of x.

• (aInc)$_\gamma$ $t\mapsto \varphi (x,t)/t^\gamma$ is almost increasing on $(0,\infty )$ with constant L.

• (Inc)$_\gamma$ $t\mapsto \varphi (x,t)/t^\gamma$ is increasing on $(0,\infty )$.

• (aDec)$_\gamma$ $t\mapsto \varphi (x,t)/t^\gamma$ is almost decreasing on $(0,\infty )$ with constant L.

• (Dec)$_\gamma$ $t\mapsto \varphi (x,t)/t^\gamma$ is decreasing on $(0,\infty )$.

• (A0) $L^{-1}\leqslant \varphi (x,1)\leqslant L$.

We write or if or holds for some $\gamma > 1$.

We can rewrite or with $p,q>0$ and constant $L\geqslant 1$ as

$$\begin{aligned} \varphi (x,\lambda t)\leqslant L\lambda ^p\varphi (x,t) \quad \text {and}\quad \varphi (x,\Lambda t) \leqslant L \Lambda ^q \varphi (x,t), \quad \text {respectively}, \end{aligned}$$

for all $(x,t)\in \Omega \times [0,\infty )$ and $0\leqslant \lambda \leqslant 1 \leqslant \Lambda$. From these inequalities one sees that and are equivalent to the $\nabla _2$- and $\Delta _2$-conditions, respectively. The definition of above differs slightly from [33] but the two definitions are equivalent when $\varphi$ satisfies . If $\varphi (x,\cdot )\in C^1((0,\infty ))$, then for $0<p \leqslant q$,

Suppose $\varphi ,\psi : [0,\infty )\rightarrow [0,\infty )$ are increasing, $\varphi$ satisfies and , and $\psi$ satisfies . Then there exist a convex ${\tilde{\varphi }}$ and a concave ${\tilde{\psi }}$ such that $\varphi \approx {\tilde{\varphi }}$ and $\psi \approx {\tilde{\psi }}$ [33, Lemma 2.2.1]. Therefore, by Jensen’s inequality for ${\tilde{\varphi }}$ and ${\tilde{\psi }}$,

for every $f\in L^1(\Omega )$ with implicit constants depending on L from and or (via the constants from the equivalence relation).

We next introduce classes of $\Phi$-functions and generalized Orlicz spaces following [33]. We are mainly interested in convex functions for minimization problems and related PDEs, but the class ${\Phi _{\textrm{w}}}(\Omega )$ is very useful for approximating functionals.

Definition 2.2

Let $\varphi :\Omega \times [0,\infty ]\rightarrow [0,\infty )$. Assume $x\mapsto \varphi (x,|f(x)|)$ is measurable for every measurable function f on $\Omega$, $t\mapsto \varphi (x,t)$ is increasing for every $x\in \Omega$, and $\varphi (x,0)=\lim _{t\rightarrow 0^+}\varphi (x,t)=0$ and $\lim _{t\rightarrow \infty }\varphi (x,t)=\infty$ for every $x\in \Omega$. Then $\varphi$ is called a

• (1) $\Phi$-function, denoted $\varphi \in {\Phi _{\textrm{w}}}(\Omega )$, if it satisfies ;

• (2) convex $\Phi$-function, denoted $\varphi \in {\Phi _{\textrm{c}}}(\Omega )$, if $t\mapsto \varphi (x,t)$ is left-continuous and convex for every $x\in \Omega$.

If ${\bar{\varphi }}$ is independent of x and $\varphi (x,t):= {\bar{\varphi }}(t)$ satisfies $\varphi \in {\Phi _{\textrm{w}}}(\Omega )$ or $\varphi \in {\Phi _{\textrm{c}}}(\Omega )$, we write ${\bar{\varphi }}\in {\Phi _{\textrm{w}}}$ or ${\bar{\varphi }}\in {\Phi _{\textrm{c}}}$.

Note that ${\Phi _{\textrm{c}}}(\Omega )\subset {\Phi _{\textrm{w}}}(\Omega )$ since convexity implies . For $\varphi ,\psi \in {\Phi _{\textrm{w}}}(\Omega )$ the relation $\simeq$ is weaker than $\approx$, but they are equivalent if $\varphi$ and $\psi$ satisfy . We write

$$\begin{aligned} \varphi ^+_{B_r}(t):=\sup _{x\in B_r\cap \Omega }\varphi (x,t) \quad \text {and}\quad \varphi ^-_{B_r}(t):=\inf _{x\in B_r\cap \Omega }\varphi (x,t). \end{aligned}$$

The (left-continuous) inverse function with respect to t is defined by

$$\begin{aligned} \varphi ^{-1}(x,t):= \inf \{\tau \geqslant 0: \varphi (x,\tau )\geqslant t\}. \end{aligned}$$

If $\varphi$ is strictly increasing and continuous in t, then this is just the normal inverse. We define the conjugate function of $\varphi \in {\Phi _{\textrm{w}}}(\Omega )$ by

$$\begin{aligned} \varphi ^*(x,t):=\sup _{s\geqslant 0} \, (st-\varphi (x,s)). \end{aligned}$$

The definition directly implies Young’s inequality

$$\begin{aligned} ts\leqslant \varphi (x,t)+\varphi ^*(x, s)\quad \text {for all }\ s,t\geqslant 0. \end{aligned}$$

If $\varphi$ satisfies or for some $p, q>1$, then $\varphi ^*$ satisfies or , respectively; the prime denotes the Hölder conjugate, $p'=\frac{p}{p-1}$. We also note that $(\varphi ^*)^*=\varphi$ if $\varphi \in {\Phi _{\textrm{c}}}(\Omega )$ by [24, Theorem 2.2.6].

If $\varphi \in {\Phi _{\textrm{c}}}(\Omega )$, then there exists an increasing and right-continuous $\varphi ':\Omega \times [0,\infty )\rightarrow [0,\infty )$ such that

$$\begin{aligned} \varphi (x,t)=\int _0^t \varphi '(x,s)\, ds. \end{aligned}$$

We collect some results about this (right-)derivative $\varphi '$.

Proposition 2.3

(Proposition 3.6, [41]) Let $\gamma >0$ and $\varphi \in {\Phi _{\textrm{c}}}(\Omega )$.

• (1) If $\varphi '$ satisfies , , or , then $\varphi$ satisfies , , or , respectively, with the same constant $L\geqslant 1$.

• (2) If $\varphi$ satisfies , then $(2^{\gamma +1}L)^{-1}t \varphi '(x,t) \leqslant \varphi (x,t)\leqslant t \varphi '(x,t)$.

• (3) If $\varphi '$ satisfies and with constant $L\geqslant 1$, then $\varphi$ also satisfies , with constant depending on L and $\gamma$.

• (4) $\varphi ^*(x,\varphi '(x,t))\leqslant t\varphi '(x,t)$.

Let $L^0(\Omega )$ be the set of the measurable functions on $\Omega$. For $\varphi \in {\Phi _{\textrm{w}}}(\Omega )$, the generalized Orlicz space (also known as the Musielak–Orlicz space) is defined as

$$\begin{aligned} L^{\varphi }(\Omega ):=\big \{f\in L^0(\Omega ):\Vert f\Vert _{L^\varphi (\Omega )}<\infty \big \}, \end{aligned}$$

with the (Luxemburg) norm

$$\begin{aligned} \Vert f\Vert _{L^\varphi (\Omega )}:=\inf \bigg \{\lambda >0: \varrho _{\varphi }\Big (\frac{f}{\lambda }\Big )\leqslant 1\bigg \}, \quad \text {where}\quad \varrho _{\varphi }(f):=\int _\Omega \varphi (x,|f|)\,dx. \end{aligned}$$

We denote by $W^{1,\varphi }(\Omega )$ the set of functions $f\in W^{1,1}(\Omega )$ with $\Vert f\Vert _{W^{1,\varphi }(\Omega )}:=\Vert f\Vert _{L^\varphi (\Omega )}+\big \Vert |Df|\big \Vert _{L^\varphi (\Omega )}<\infty$. Note that if $\varphi$ satisfies , then $f\in L^\varphi (\Omega )$ if and only if $\varrho _\varphi (f)<\infty$, and if $\varphi$ satisfies , and , then $L^\varphi (\Omega )$ and $W^{1,\varphi }(\Omega )$ are reflexive Banach spaces. We denote by $W^{1,\varphi }_0(\Omega )$ the closure of $C^\infty _0(\Omega )$ in $W^{1,\varphi }(\Omega )$. For more information about generalized Orlicz and Orlicz–Sobolev spaces, we refer to the monographs [14, 33] and also [24, Chapter 2].

3 Lower regularity with a priori assumptions

3.1 Continuity assumptions

The condition , introduced in [39] (see also [44]), is a “almost continuity” assumption, which allows the function to jump, but not too much. It implies the Hölder continuity of solutions and (quasi)minimizers [11, 36, 37]. For higher regularity, we introduced in [41] a “vanishing ” condition, denoted , and a weak vanishing version, and generalized them to the quasi-isotropic situation in [42]. The anisotropic condition was further studied in [13, 40]. These previous studies applied to the “natural” energy assumption $u\in W^{1,\varphi }(\Omega )$, called Case (ap1) in the introduction. The and conditions for a priori energy assumptions were developed in [36] and [11] for functions in $L^\infty$ and $W^{1,\psi }$, respectively. Here we generalize and unify all the conditions for a priori information; the most important one for this article is .

Definition 3.1

Let $M,N\in {\mathbb {N}}$, $G:\Omega \times \mathbb {R} ^M \rightarrow \mathbb {R} ^N$, $\psi :\Omega \times \mathbb {R} ^M\rightarrow [0,\infty )$, $L_\omega >0$, $r\in (0,1]$ and $\omega :[0,1]\rightarrow [0,L_\omega ]$. We consider the claim

$$\begin{aligned} |G(x,\xi )-G(y,\xi )|\leqslant \omega (r)\big (|G(y,\xi )|+1\big ) \quad \text {when }\ \psi (y,\xi )\in [0,|B_r|^{-1}] \end{aligned}$$

for all $x,y\in B_r\cap \Omega$ and $\xi \in \mathbb {R} ^M$. We say that G satisfies:

• (A1-$\psi$) if there exists $L_\omega$ such that the claim holds with $\omega \equiv L_\omega$.

• (VA1-$\psi$) if there exists $L_\omega$ and a modulus of continuity $\omega$ such that the claim holds.

• (wVA1-$\psi$) if it satisfies for every $\varepsilon >0$, with possibly different functions $\omega _\varepsilon$ but a common $L_\omega$ independent of $\varepsilon$.

When $\psi (x,\xi )=|\xi |^s$ with $s>0$ we use the abbreviations , and and in the case $\psi =|G|$ we write , and . We also use the definition for $\psi :\Omega \times [0,\infty )\rightarrow [0,\infty )$ with the understanding that $\psi (x,\xi )=\psi (x,|\xi |)$.

Table 1 Examples of sufficient conditions in special cases

It can be easily seen that $\Longrightarrow$$\Longrightarrow$ and $\Longrightarrow$ if $s'\leqslant s$, similarly for and . In the case $M=N=1$, these conditions are somewhat differently formulated than in earlier papers, but we showed in [42] that the formulations are are equivalent to previous versions under natural assumptions on G.

The next example shows the relevance of the parameter s in in the double phase case. Similar relations for other cases are summarized in Table 1.

Example 3.2

Consider two double phase energies for $1<p \leqslant q$ and $a:\Omega \rightarrow [0,L]$.

• Let $\varphi _1(x, t):= t^p + a(x)t^q$, $a\in VC^{0,\alpha }(\Omega )$ for some $\alpha \in (0,1]$.

  If $q-p \leqslant \tfrac{s}{n} \alpha$, then $\varphi _1$ satisfies with $\omega$ proportional to the modulus of continuity of a.

• Let $\varphi _2(x, t):= t^p + a(x)^\alpha t^q$, where $\alpha >0$ and $a\in C^{0,1}(\Omega )$.

  If $q-p\leqslant \tfrac{s}{n} \alpha$, then $\varphi _2$ satisfies and with $\omega _\varepsilon (r) = c r^{\{\alpha -\frac{n(q-p)}{(1+\varepsilon )s}\}\min \{1,\frac{1}{\alpha }\}}$;

  If $q-p< \tfrac{s}{n} \alpha$, then $\varphi _2$ satisfies with $\omega (r) = c r^{(\alpha -\frac{n(q-p)}{s})\min \{1,\frac{1}{\alpha }\}}$.

Note that these conditions can hold for $\frac{q}{p}$ arbitrarily large. We show the second case only since the first case can be obtained in the same way as the second case with $\alpha <1$. We will use the elementary inequality

$$\begin{aligned} |a^\alpha -b^\alpha | \leqslant {\left\{ \begin{array}{ll} |a-b|^\alpha , &{} 0< \alpha \leqslant 1,\\ c_\alpha \delta ^{1-\alpha } |a-b|^\alpha +\delta b^\alpha , &{}\quad \alpha >1, \end{array}\right. } \end{aligned}$$

which holds for any $a, b \geqslant 0$ and $\delta \in (0,1]$; here $c_\alpha >0$ is a constant depending on $\alpha$. The second inequality ($\alpha >1$) follows from Young’s inequality applied to the right-hand side of $b^\alpha -a^\alpha \leqslant \alpha b^{\alpha -1}(b-a)$ when $b\geqslant a \geqslant 0$.

Suppose that $q-p\leqslant \frac{s}{n}\alpha$ and let $x,y\in B_r$ with $r\in (0,1]$ and $t \in (0,|B_r|^{-\frac{1}{s(1+\varepsilon )}}]$ with $\varepsilon \geqslant 0$. Applying the preceding inequality with $a=a(x)$ and $b=a(y)$, we obtain that

$$\begin{aligned} |\varphi _2(x,t)-\varphi _2(y,t)| \lesssim r^\alpha t^q = r^\alpha t^{q-p} t^p \lesssim r^{\alpha -\frac{n(q-p)}{(1+\varepsilon )s}} \varphi _2(y,t) \end{aligned}$$

when $0<\alpha \leqslant 1$; in the case $\alpha > 1$, we choose $\delta :=r^{\{\alpha -\frac{n(q-p)}{(1+\varepsilon )s}\}\frac{1}{\alpha }}$ and find that

$$\begin{aligned}\begin{aligned} |\varphi _2(x,t)-\varphi _2(y,t)|&\lesssim (\delta ^{1-\alpha } r^\alpha +\delta a(y)^\alpha ) t^q\\&\lesssim \delta ^{1-\alpha } r^{\alpha -\frac{n(q-p)}{(1+\varepsilon )s}} t^p +\delta a(y)^\alpha t^q = r^{\{\alpha -\frac{n(q-p)}{(1+\varepsilon )s}\}\frac{1}{\alpha }}\varphi _2(y,t). \end{aligned}\end{aligned}$$

These inequalities imply the desired , and -conditions.

Let us show how to use the smallness of $\omega$ to obtain the inequality from for a slightly larger range. Intuitively, we shift some power from the coefficient to the range. In this proof it is important that the range of $\xi$ in the condition is independent of x, so the result does not generalize to easily, unless $\psi (x,t)=\psi (t)$.

Proposition 3.3

Let $G:\Omega \times \mathbb {R} ^M \rightarrow \mathbb {R} ^N$ satisfy , $\theta \in [0,1]$ and $r\in (0,1]$. If $\omega (r)^{1-\theta }\leqslant \tfrac{1}{3}$, then for every $x,y \in B_r\cap \Omega$,

$$\begin{aligned} |G(x,\xi )-G(y,\xi )|\leqslant \omega (r)^\theta \big (|G(y,\xi )|+1\big ) \quad \text {when }\ 3^n\omega (r)^{n(1-\theta )} |\xi |^s\in [0,|B_r|^{-1}]. \end{aligned}$$

Proof

Note by the concavity of $\log$ that $t\log 2\leqslant \log (1+t)\leqslant t$ when $t\in [0,1]$, and set

$$\begin{aligned} k:=\Big \lfloor \frac{\log (1+\omega (r)^\theta )}{\log (1+\omega (r))}\Big \rfloor \geqslant \big \lfloor \omega (r)^{\theta -1} \log 2 \big \rfloor \geqslant \frac{\log 2}{2} \omega (r)^{\theta -1} > \frac{1}{3} \omega (r)^{\theta -1} \geqslant 1, \end{aligned}$$

where $\lfloor \tau \rfloor$ is the largest integer less than or equal to $\tau \in \mathbb {R}$. Suppose $x,y\in B_r\cap \Omega$ and $3^n\omega (r)^{n(1-\theta )} |\xi |^s \leqslant |B_r|^{-1}$. Then $|\xi |^s\leqslant k^n |B_r|^{-1}$ since $1\leqslant (3k)^n\omega (r)^{n(1-\theta )}$. We split the segment [x, y] into k equally long subsegments $[x_i, x_{i+1}]$ with $x_0=x$ and $x_k=y$ so that $x_i, x_{i+1}\in B_{r/k}$. Since $|\xi |^s\leqslant |B_{r/k}|^{-1}$, we can use to estimate

$$\begin{aligned} \begin{aligned} |G(x_i, \xi )|+1&\leqslant (1+\omega (r))(|G(x_{i+1}, \xi )| + 1) \leqslant \cdots \leqslant (1+\omega (r))^{k-i}(|G(y, \xi )| + 1). \end{aligned} \end{aligned}$$

We use this estimate with the triangle inequality and :

$$\begin{aligned} \begin{aligned} |G(x,\xi )-G(y,\xi )|&\leqslant \sum _{i=0}^{k-1} |G(x_{i+1},\xi )-G(x_i,\xi )| \leqslant \omega (r)\sum _{i=1}^{k} (|G(x_i,\xi )| +1) \\&\leqslant \omega (r) \sum _{i=1}^{k} (1+\omega (r))^{k-i} (|G(y,\xi )| +1)\\&= [(1+\omega (r))^k -1](|G(y,\xi )| +1). \end{aligned} \end{aligned}$$

This gives the desired estimate, since by the definition of k,

$$\begin{aligned} (1+\omega (r))^k \leqslant (1+\omega (r))^{\frac{\log (1+\omega (r)^\theta )}{\log (1+\omega (r))}} = 1+\omega (r)^\theta . \end{aligned}$$

$\square$

3.2 Sobolev–Poincaré inequality

We derive a modular Sobolev–Poincaré-type inequality in generalized Orlicz spaces assuming a priori information. We first state the inequality with an abstract condition, which is explored further in Lemma 3.8 and Example 3.9. The example shows that the conditions in Lemma 3.8 are essentially sharp for the Sobolev–Poincaré inequality, at least when $s\leqslant n$. This approach is inspired by [11].

Theorem 3.4

(Sobolev–Poincaré inequality) Let $\varphi \in {\Phi _{\textrm{w}}}(B_r)$ satisfy , and with $1\leqslant p\leqslant q$, and let $u\in W^{1,1}(B_r)$. If

(3.5)

and some $b_0,\theta >0$, then

for $\frac{1}{\theta _0} = 1-\min \{\frac{p}{n},\kappa \}+\frac{1}{\theta }$ with any $\kappa \in (0,1)$ and some $c=c(n,p,q,L,\kappa ,b_0)>0$.

Remark 3.6

In the previous theorem we can choose $\theta _0>1$ if and only if $\theta >\max \{1,\frac{n}{p}\}$. The choice $\theta _0=1$ is additionally possible when $\theta =\frac{n}{p} > 1$. These are the most important cases, but the theorem allows also for $\theta _0\in (0,1)$ in cases with small $\theta$.

Remark 3.7

Let $1<p \leqslant q$. Suppose that $\varphi$ satisfies (3.5) with $\theta$ such that $\theta _0>1$. Then so does $\varphi ^{1/\tau }$ with $\theta ':=\theta \tau$ and $\tau \in (1,p)$. Theorem 3.4 for $\varphi ^{1/\tau }$ implies that

where $\frac{1}{\theta _0'} = 1-\min \{\frac{p}{n},\kappa \}+\frac{1}{\theta \tau }$. Note that $\theta _0'\rightarrow \theta _0>1$ when $\tau \rightarrow 1$ so we can choose $\tau$ with $\frac{\theta _0'}{\tau }>1$. Thus there exists $\theta _1>1$ depending n, p and $\theta$ such that

Proof of Theorem 3.4

To obtain a differentiable function, we define $\psi \in {\Phi _{\textrm{c}}}$ by

$$\begin{aligned} \psi (t):= \int _0^t \sup _{\sigma \in (0,\tau ]} \frac{\varphi ^-_{B_r}(\sigma )}{\sigma }\, d\tau . \end{aligned}$$

From of $\varphi$ we see that $\frac{\varphi ^-_{B_r}(t)}{t} \leqslant \psi '(t) \leqslant L \frac{\varphi ^-_{B_r}(t)}{t}$; with and we conclude that $\varphi _{B_r}^-\approx \psi (t)$. Choose $s:=\min \{p,\kappa n\}\in [1,n)$ and $\theta _0\in (0, \theta )$ with $\frac{s}{n} = 1-\frac{1}{\theta _0}+\frac{1}{\theta }$. Note that $\frac{s^*}{\theta _0s}=((1-\frac{s}{n})\theta _0)^{-1}=(1-\frac{\theta _0}{\theta })^{-1}>1$. By Hölder’s inequality with exponents $\frac{s^*}{\theta _0s}$ and $(\frac{s^*}{\theta _0s})'=\frac{\theta }{\theta _0}$ and the assumption (3.5),

We use the Sobolev–Poincaré inequality in $L^s$ and $\psi '(t)\approx \psi (t)/t$ to conclude that

Since $\psi$ satisfies and $s\leqslant p$, $\psi _s(t):=\psi (t^{1/s})$ satisfies . Therefore Young’s inequality for $\psi _s$ and $\psi _s^*(\frac{\psi _s(t)}{Lt})\leqslant \psi _s(t)$ [35, Lemma 3.1] with $t:=v^s$ give $\frac{\psi (v)}{v^s} |\nabla u|^s \lesssim \psi (v) + \psi (|\nabla u|)$. Continuing the previous estimate with the $L^{s^*}$-triangle inequality and $\psi \approx \varphi ^-_{B_r}$, we find that

By Hölder’s inequality, $(\psi (v)^{\frac{1}{s}})_{B_r}^s\leqslant (\psi (v))_{B_r}$ and by the modular Poincaré inequality in the Orlicz space $L^\psi$ [33, Corollary 7.4.1], $(\psi (v))_{B_r}$ can be estimated by the first term on the right-hand side. Combined with the inequality from the previous paragraph, this gives the claim. $\square$

Let us derive some sufficient conditions for the assumption of the previous theorem by complementing [11, Proposition 4.2]. Also the cases $u\in L^{\psi ^\#}(B_r)$ and $u\in W^{1,\psi }(B_r)$ for $\psi \in {\Phi _{\textrm{w}}}(B_r)$ and are covered by [11], and could likewise be considered here. We define the bounded mean oscillation semi-norm as

Note that in case (3) of the following lemma we need $s>n(1-\frac{p}{q})$ in order that $\theta _0>1$ in Theorem 3.4, cf. Remark 3.6.

Lemma 3.8

Let $\varphi \in {\Phi _{\textrm{w}}}(B_r)$ satisfy , , and with $1\leqslant p \leqslant q$ and let $u\in L^1(B_r)$. Assume one of the following holds:

• (1) $s>n$ and $[u]_{\gamma , B_r} \leqslant b$ with $\gamma := 1-\frac{n}{s}$.

• (2) $s=n$ and $[u]_{BMO(B_r)}\leqslant b$.

• (3) $s\in [1, n)$ and $\Vert u\Vert _{L^{s^*}(B_r)} \leqslant b$.

Then (3.5) holds for any $\theta >0$ in Cases (1)–(2) and for $\theta =\frac{s^*}{q-p}$ in Case (3). The constant $b_0$ depends only on n, p, q, L, $L_\omega$ and b.

Proof

If $[u]_{\gamma }\leqslant b$, then $v\leqslant 2|B_1|^{\frac{1}{n}} b |B_r|^{\frac{\gamma -1}{n}}$ and $\varphi ^+_{B_r}(v)\lesssim \varphi ^-_{B_r}(v)+1$ by , so the claim holds in Case (1). For the same reason and , the integrand in (3.5) in Case (2) is bounded at points with $v\leqslant \max \{|B_r|^{-1/n},1\}$. On the other hand, at points with $v>\max \{|B_r|^{-1/n},1\}$ we estimate, by , , and ,

$$\begin{aligned} \frac{\varphi (x,v)}{\varphi _{B_r}^-(v)+1} \approx \frac{\varphi (x,v)}{\varphi _{B_r}^-(v)} \lesssim \Big (\frac{v}{|B_r|^{-1/n}}\Big )^{q-p} \frac{\varphi (x,|B_r|^{-1/n})}{\varphi _{B_r}^-(|B_r|^{-1/n})} \approx \Big (\frac{v}{r^{-1}}\Big )^{q-p} = |u-(u)_{B_r}|^{q-p}. \end{aligned}$$

We obtain for any exponent $\theta >0$ that

where in the last inequality we use the well-known reverse Hölder type inequality for mean oscillations in $L^{\theta (q-p)}$-space and BMO (cf. [26, Lemma A.1]). Case (3) was proved in [11, Proposition 4.2]. $\square$

The estimates for the Sobolev–Poincaré inequality may seem crude, but the following example shows that the end result is sharp, i.e. the claim is false if is replaced by for any $s'<s$. See also [11, Section 5] for a one-dimensional example.

Example 3.9

Let $n=2$ and denote the quadrants by $Q_k\subset \mathbb {R} ^2$, $k\in \{1,2,3,4\}$. Let $\eta :[0,4]\rightarrow [0,1]$ be the piecewise linear, 3-Lipschitz function with $\eta _{[\frac{1}{3}, \frac{2}{3}]}=\eta _{[\frac{7}{3}, \frac{8}{3}]}=1$ and $\eta _{[1,2]}=\eta _{[3,4]}=0$. We define $a:\mathbb {R} ^2\rightarrow [0,\infty )$ in polar coordinates as $a(r,\theta ):=r^\alpha \eta (\frac{2}{\pi }\theta )$. Thus a equals 0 in $Q_2$ and $Q_4$ and $a(x)=|x|^\alpha$ in the sectors with $\frac{\pi }{6}<\theta <\frac{\pi }{3}$ in $Q_1$ and with $\frac{7\pi }{6}<\theta <\frac{4\pi }{3}$ in $Q_3$. Consider the double phase functional $H(x,t)=t^p + a(x)t^q$ with $p<2$ and the function $u:\mathbb {R} ^2\rightarrow \mathbb {R}$ which equals 1 in $Q_1$, $-1$ in $Q_3$ and is linear in the polar coordinate $\theta$ in $Q_2$ and $Q_4$. By symmetry, $u_{B_r}=0$ for every ball $B_r$ centered at the origin and $v:=\frac{1}{r} |u-(u)_{B_r}| = \frac{1}{r}$ in the sectors in $Q_1$ and $Q_3$. The derivative of u equals zero in $Q_1$ and $Q_3$; in the other quadrants the radial derivative is zero, and in the tangential derivative equals $\frac{4}{\pi r}$. For a constant $k>0$ we estimate, based on the sectors in $Q_1$ and $Q_3$,

and, since the support of the derivative is $Q_2 \cup Q_4$ where $a=0$,

If the modular Poincaré inequality from Theorem 3.4 holds with $\theta _0=1$ (the weakest relevant case), then

$$\begin{aligned} r^{\alpha -q} k^q \leqslant c r^{-p}k^p + c \quad \text {so that} \quad r^{\alpha -(q-p)} k^{q-p} \leqslant c + cr^{p}k^{-p}. \end{aligned}$$

Suppose that we want the constant in the inequality to depend on the $L^{s^*}$-norm of ku. We calculate $\Vert ku\Vert _{L^{s^*}(B_r)}=ckr^{2/s^*}$. Thus $k = c r^{-2/s^*}$ and so

$$\begin{aligned} r^{\alpha -(q-p)} k^{q-p} \approx r^{\alpha - (\frac{2}{s^*}+1)(q-p)} = r^{\alpha - \frac{2}{s}(q-p)}. \end{aligned}$$

This remains bounded as $r\rightarrow 0$ when $\alpha \geqslant \frac{2}{s}(q-p)$ which is exactly the condition when $n=2$ and shows the sharpness of Case (3). If $s=n=2$, this shows the sharpness of Case (2), even if we allow the constant to depend on the $L^\infty$-norm. Similarly, we see that if the constant is allowed to depend on $\rho _H(|\nabla u|)$, then $\alpha \geqslant \frac{2}{p}(q-p)$ which is . Unfortunately, Case (1) is not covered, since the counter-example is discontinuous.

3.3 Quasiminimizers

In this subsection, we derive regularity results for quasiminimizers with a priori information. Let $\varphi \in {\Phi _{\textrm{w}}}(\Omega )$. We say that $u\in W^{1,\varphi }_{\textrm{loc}}(\Omega )$ is a (local) quasiminimizer if there exists $Q\geqslant 1$ such that

$$\begin{aligned} \int _{{\text {supp}}\,(u-v)} \varphi (x,|Du|)\,dx \leqslant Q \int _{{\text {supp}}\,(u-v)} \varphi (x,|Dv|)\,dx \end{aligned}$$

for every $v\in W^{1,\varphi }_{\textrm{loc}}(\Omega )$ with ${\text {supp}}\,(u-v)\Subset \Omega$. Quasiminimizers of energy functionals with generalized Orlicz growth have been studied e.g. in [11, 12, 36,37,38]. If $\varphi$ satisfies , then the quasiminimizer u satisfies the Caccioppoli inequality

$$\begin{aligned} \int _{B_r} \varphi (x,|Du|)\, dx \leqslant c \int _{B_{2r}} \varphi \bigg (x,\frac{|u-(u)_{B_{2r}}|}{r}\bigg )\, dx, \end{aligned}$$

(3.10)

for some $c=c(n,q,L,Q)\geqslant 1$ and every $B_{2r}\Subset \Omega$, see [36, Lemma 4.6].

If $\varphi$ satisfies , then a bounded qusiminimizer satisfies a Harnack-type inequality and so is locally Hölder continuous [36, Theorem 4.1]. The main ingredients of the proof are the Caccioppoli estimate (3.10) and the Sobolev–Poincaré inequality (Theorem 3.4). In Lemma 3.8, we derived several sufficient conditions for the Sobolev–Poincaré inequality. Therefore, we obtain the Hölder continuity under these conditions from almost the same proof as [36, Theorem 4.1]. We start with local boundedness of quaisiminimizers. The proof is exactly the same as [36, Proposition 5.5] and is hence omitted.

Lemma 3.11

Let $\varphi \in {\Phi _{\textrm{w}}}(\Omega )$ satisfy , , and with $1<p \leqslant q$, and $s\in (1,n]$. Assume that $u\in W^{1,\varphi }_{\textrm{loc}}(\Omega )$ is a quasiminimizer and $u\in BMO(\Omega )$ with $s=n$, or $u\in L^{s^*}(\Omega )$ with $s\in (n(1-\frac{p}{q}), n)$. Then $u\in L^{\infty }_{\textrm{loc}}(\Omega )$.

Theorem 3.12

Let $\varphi \in {\Phi _{\textrm{w}}}(\Omega )$ satisfy , , and with $1<p \leqslant q$, and $s\in (1,n]$. Assume that $u\in W^{1,\varphi }_{\textrm{loc}}(\Omega )$ is a quasiminimizer and one of the following holds:

• (1) $s=n$ and $u\in BMO(\Omega )$.

• (2) $s\in (n(1-\frac{p}{q}), n)$ and $u\in L^{s^*}(\Omega )$.

For every $\Omega '\Subset \Omega$ there exists $\gamma \in (0,1)$ depending only on n, p, q, L, $L_\omega$, Q and $\Omega '$ such that $u\in C^{0,\gamma }(\Omega ')$.

Proof

Assume first that $\varphi$ satisfies and $u\in W^{1,\varphi }_{\textrm{loc}}(\Omega )\cap L^\infty (\Omega )$. Then local Hölder continuity follows directly from the Harnack inequality in [36, Theorem 4.1]. We consider then assumptions (1) and (2) and note that Lemma 3.11 implies that $u\in L^{\infty }_{\textrm{loc}}(\Omega )$. Furthermore, implies when $s<n$. Therefore, we obtain the local Hölder continuity from the result for bounded solutions. $\square$

We end the section with a higher integrability result for Hölder continuous quasiminimizers. We first observe that if $u\in C^{0,\gamma }(B_{2r})$ for some $\gamma \in (0,1)$ and $\varphi$ satisfies , then by Jensen’s inequality and the Caccioppoli inequality (3.10)

which implies

(3.13)

Here c depends on n, p, q, L, $L_\omega$ and Q, and is independent of $\gamma$.

Theorem 3.14

(Reverse Hölder inequality) Let $\varphi \in {\Phi _{\textrm{w}}}(\Omega )$ satisfy , , and with $1<p\leqslant q$ and $\gamma \in (0,1)$. If $u\in W^{1,\varphi }_{\textrm{loc}}(\Omega )\cap C^{0,\gamma }(B_{4r})$ is a quasiminimizer in $B_{4r}\Subset \Omega$, then $|Du|\in L^{\varphi ^{1+\sigma }}(B_r)$ for some $\sigma >0$ with the estimate

The constants $\sigma$ and c depend only on n, p, q, L, $L_\omega$, Q and $[u]_{\gamma , B_{2r}}$.

Proof

By the Caccioppoli estimate (3.10) and the Sobolev–Poincaré inequality (from Remark 3.7 and Lemma 3.8(1)), we obtain

for every ball $B_{2\rho }\subset B_{2r}$, where $\theta _1$ and $c_1$ depend on the parameters listed in the statement. By Gehring’s Lemma (e.g. [31, Theorem 6.6]), there exists $\sigma >0$ depending on $\theta _1$ and $c_1$ such that

for every ball $B_{2\rho }\subset B_{2r}$. Moreover, using the technique from [41, Lemma 4.7] with instead of and with (3.13) in $B_\rho$, we obtain for every ball $B_{2\rho }\subset B_{2r}$ that

$\square$

4 Growth functions and autonomous problems

Let us precisely define our solutions and minimizers. Since we only consider local versions we will drop the word “local” later on, as indicated by the parentheses. We say that $u\in W^{1,1}_{\textrm{loc}}(\Omega )$ is a (local) weak solution to (${\text {div}}A$) if $|Du|\,|A(\cdot ,Du)|\in L^1_{\textrm{loc}}(\Omega )$ and

$$\begin{aligned} \int _{\Omega } A(x,Du)\cdot D\zeta \,dx = 0 \end{aligned}$$

for all $\zeta \in W^{1,1}(\Omega )$ with ${\text {supp}}\zeta \Subset \Omega$ and $|D\zeta |\,|A(\cdot , D\zeta )| \in L^1(\Omega )$. We say that $u\in W^{1,1}_{\textrm{loc}}(\Omega )$ is a (local) minimizer if $F(\cdot ,Du)\in L^1_{\textrm{loc}}(\Omega )$ and

$$\begin{aligned} \int _{{\text {supp}}(u-v)} F(x,Du)\, dx \leqslant \int _{{\text {supp}}(u-v)} F(x,Dv)\, dx \end{aligned}$$

for every $v\in W^{1,1}_{\textrm{loc}}(\Omega )$ with ${\text {supp}}(u-v)\Subset \Omega$. Note that if (${\text {div}}A$) is an Euler–Lagrange equation, that is, if $A=D_\xi F$ for some a function F, then the weak solution to (${\text {div}}A$) is a minimizer of ($\min F$).

We introduce fundamental assumptions on $A:\Omega \times {\mathbb {R}^n} \rightarrow {\mathbb {R}^n}$ or $F:\Omega \times {\mathbb {R}^n} \rightarrow [0,\infty )$ with respect to the gradient variable $\xi$ from [41, 42], so-called (p, q)-growth and quasi-isotropy conditions (parts (Aii) and (Aiii) of the definition, respectively). Here, “quasi-isotropic” indicates that A or F can be estimated by a non-autonomous isotropic $\Phi$-function, the so-called growth function.

Definition 4.1

We say that $A:\Omega \times \mathbb {R} ^n\rightarrow \mathbb {R} ^n$ or $F:\Omega \times \mathbb {R} ^n\rightarrow [0,\infty )$ has quasi-isotropic (p, q)-growth if conditions (Ai)–(Aiii) or (Fi)–(Fii) hold, respectively.

• (Ai) For every $x\in \Omega$, $A(x, 0)= 0$ and $A(x,\cdot )\in C^{1}(\mathbb {R} ^n{\setminus }\{ 0\}; \mathbb {R} ^n)$ and for every $\xi \in \mathbb {R} ^n$, $A(\cdot ,\xi )$ is measurable.

• (Aii) There exist $L\geqslant 1$ and $1<p\leqslant q$ such that the radial function $t\mapsto | D_\xi A(x,te)|$ satisfies , and with the constant L, for every $x\in \Omega$ and $e\in \partial B_1(0)$.

• (Aiii) There exists $L\geqslant 1$ such that

  $$\begin{aligned} | D_\xi A(x,\xi ')|\,|{\tilde{\xi }}|^2 \leqslant L\, D_\xi A(x,\xi ){\tilde{\xi }} \cdot {\tilde{\xi }} \end{aligned}$$

  for all $x\in \Omega$, $\xi ,\xi ', {\tilde{\xi }}\in {\mathbb {R}^n} {\setminus }\{0\}$ with $|\xi |=|\xi '|$.

• (Fi) For every $x\in \Omega$, $F(x, 0)=|D_\xi F(x,0)|= 0$ and $F(x,\cdot )\in C^{2}(\mathbb {R} ^n{\setminus }\{ 0\})$ and for every $\xi \in \mathbb {R} ^n$, $F(\cdot ,\xi )$ is measurable.

• (Fii) The derivative $A:=D_\xi F$ satisfies conditions (Aii) and (Aiii).

The assumptions (Aii) and (Aiii) for $A:=D_\xi F$ impose the two crucial conditions on the Hessian matrix $D^2_\xi F$. The former means that $|\xi |^2 D^2_{\xi }F(x,\xi )$ satisfies a (p, q)-growth condition which is a variant of (p, q)-growth of F. The latter is equivalent to the existence of ${\tilde{L}}\geqslant 1$ such that

$$\begin{aligned} \frac{\sup \{\text {eigenvalue of }D^2_\xi F(x,\xi ):\, \xi \in \partial B_t(0)\}}{\inf \{\text {eigenvalue of }D^2_\xi F(x,\xi ):\, \xi \in \partial B_t(0)\}}\leqslant {\tilde{L}} \quad \text {for all }\ x\in \Omega \ \text { and } \ t>0. \end{aligned}$$

(4.2)

Note that all examples in Table 1 satisfy this condition, but not

$$\begin{aligned} \frac{\sup \{\text {eigenvalue of }D^2_\xi F(x,\xi ):\, x\in \Omega , \xi \in \partial B_t(0)\}}{\inf \{\text {eigenvalue of }D^2_\xi F(x,\xi ):\, x\in \Omega , \xi \in \partial B_t(0)\}} \leqslant {\tilde{L}} \quad \text {for all } \ t>0, \end{aligned}$$

(4.3)

which holds for $F(x,\xi )\approx a(x)\psi (|\xi |)$ with $0<\nu \leqslant a \leqslant L$. We remark (4.2) and (4.3) are called the pointwise and global uniform ellipticity condition. Here, “uniform” is concerned with the variable $|\xi |$. For more discussion about this uniform ellipticity condition, we refer to [22].

Definition 4.4

Let $A:\Omega \times {\mathbb {R}^n} \rightarrow {\mathbb {R}^n}$ or $F:\Omega \times \mathbb {R} \rightarrow [0,\infty )$ have quasi-isotropic (p, q)-growth. We say that $\varphi \in {\Phi _{\textrm{c}}}(\Omega )$ is its growth function if there exist $1<p_1\leqslant q_1$ and $0<\nu \leqslant \Lambda$ such that $\varphi (x,\cdot )\in C^1([0,\infty ))$ for every $x\in \Omega$ and $\varphi '$ satisfies , and as well as

$$\begin{aligned} |A(x,\xi )|+ |\xi |\,|D_\xi A(x,\xi )|\leqslant \Lambda \varphi '(x,|\xi |) \quad \text {and}\quad D_\xi A(x,\xi ){\tilde{\xi }} \cdot {\tilde{\xi }} \geqslant \nu \frac{\varphi '(x,|\xi |)}{|\xi |}|{\tilde{\xi }}|^2 \end{aligned}$$

for all $x\in \Omega$ and $\xi , {\tilde{\xi }}\in {\mathbb {R}^n} {\setminus }\{ 0\}$; in the case of F we assume that the the inequalities hold for $A:=D_\xi F$.

In this paper we always use the growth function from the following proposition. Thus the additional parameters $p_1$, $q_1$, $\nu$ and $\Lambda$ only depend on the original parameters p, q and L.

Proposition 4.5

(Proposition 3.3, [42]) Every $A:\Omega \times {\mathbb {R}^n} \rightarrow {\mathbb {R}^n}$ and $F:\Omega \times \mathbb {R} \rightarrow [0,\infty )$ with quasi-isotropic (p, q)-growth has a growth function $\varphi \in {\Phi _{\textrm{c}}}(\Omega )$ with $p_1=p$, $q_1\geqslant q$, $\nu$ and $\Lambda$ depending only on p, q and L.

By Proposition 2.3, the growth function $\varphi$ satisfies , , as well as $\varphi ^*(x,\varphi '(x,t))\leqslant \varphi '(x,t)t\approx \varphi (x,t)$. Furthemore, by [42, Remark 3.4] we have the strict monotonicity condition

$$\begin{aligned} (A(x,\xi )-A(x,{\tilde{\xi }})) \cdot (\xi -{\tilde{\xi }}) \gtrsim \frac{\varphi '(x,|\xi |+|{\tilde{\xi }}|)}{|\xi |+|{\tilde{\xi }}|}|\xi - {\tilde{\xi }}|^2, \qquad x\in \Omega , \ \ \xi ,{\tilde{\xi }}\in {\mathbb {R}^n} \setminus \{0\},\nonumber \\ \end{aligned}$$

(4.6)

as well as the equivalences

$$\begin{aligned} \varphi (x,|\xi |) \approx A(x,\xi )\cdot \xi \approx |\xi | |A(x,\xi )| \quad \text {and}\quad F(x,\xi )\approx \varphi (x,|\xi |), \qquad x\in \Omega ,\ \ \xi \in {\mathbb {R}^n} ,\nonumber \\ \end{aligned}$$

(4.7)

where the implicit constants depend on only p, q and L.

Let $A:\Omega \times {\mathbb {R}^n} \rightarrow {\mathbb {R}^n}$ have quasi-isotropic (p, q)-growth and growth function $\varphi$ and let $u\in W^{1,1}_{\textrm{loc}}(\Omega )$ be a weak solution to (${\text {div}}A$). We showed in [42, Section 4.1] that u is a quasiminimizer of the $\varphi$-energy for some $Q=Q(p,q,L)\geqslant 1$; in the reference we assumed that smooth functions are dense in the Sobolev space $W^{1,\varphi }$, which is reasonable if $\varphi$ satisfies . But this is not needed here due to our changed test function class in the definition of weak solution. Thus we can apply the regularity results for quasiminimizers from Sect. 3 to weak solutions.

We next show how the -type condition of A or F transfers to the growth function $\varphi$. Based on Part (2), we can say that and are weaker in the minimization case than in the PDE case. This justifies studying minimizers separately.

Proposition 4.8

Let $A:\Omega \times {\mathbb {R}^n} \rightarrow {\mathbb {R}^n}$ or $F:\Omega \times \mathbb {R} \rightarrow [0,\infty )$ have quasi-isotropic (p, q)-growth and growth function $\varphi$, and let $\psi :\Omega \times {\mathbb {R}^n} \rightarrow [0,\infty )$.

• (1) A or F satisfies if and only if $\varphi$ satisfies .

• (2) If $A:= D_\xi F$ satisfies , then so does F, with the same $\omega$ up to a constant depending on p, q and L.

• (3) If A or F satisfies , $s>0$, and $\theta \in (0,1]$, then

  $$\begin{aligned} |A(x,\xi )-A(y,\xi )| \leqslant c\, \omega (r)^\theta \big (\varphi '(y,|\xi |)+1\big ) \end{aligned}$$

  or

  $$\begin{aligned} |F(x,\xi )-F(y,\xi )| \leqslant c\, \omega (r)^\theta \big (\varphi (y,|\xi |)+1\big ) \end{aligned}$$

  for $B_r$ with $r\in (0,1]$, $x,y\in B_r\cap \Omega$ and $\xi \in \mathbb {R} ^n$ with $3^n\omega (r)^{n(1-\theta )} |\xi |^s\in [0,|B_r|^{-1}]$. The constant $c>0$ depends only on p, q and L.

Proof

We present only the proofs for A, as the ones for F are similar but simpler. All implicit constants in the proof depend only on p, q and L. Fix $x,y\in B_r\cap \Omega$ and $\xi \in {\mathbb {R}^n}$ with $\psi (y,\xi )\in [0,|B_r|^{-1}]$.

We first prove (1). Suppose that A satisfies and abbreviate $t:=|\xi |$. By of A and the equivalence (4.7),

$$\begin{aligned}\begin{aligned} \varphi (x,t)&\approx t |A(x,\xi )| \leqslant t |A(x,\xi )-A(y,\xi )| +t |A(y,\xi )| \\&\lesssim t (|A(y,\xi )| + 1) + \varphi (y,t) \lesssim \varphi (y,t) + t \lesssim \varphi (y,t)+1; \end{aligned}\end{aligned}$$

in the last inequality we used and of $\varphi$ when $t>1$. Thus $\varphi$ satisfies . Conversely, suppose $\varphi$ satisfies . Then we see, using also and , that

$$\begin{aligned} |A(x,\xi )| \approx \frac{\varphi (x,|\xi |)}{|\xi |} \lesssim {\left\{ \begin{array}{ll} 1 \quad \text {if }\ |\xi |\leqslant 1, \\ \frac{\varphi (y,|\xi |)+1}{|\xi |} \lesssim \frac{\varphi (y,|\xi |)}{|\xi |} \approx |A(y,\xi )| \quad \text {if }\ |\xi | > 1. \end{array}\right. } \end{aligned}$$

Hence A satisfies .

We omit the proof of (2) which is essentially the same as [42, Proposition 3.8].

To prove (3), we note by and Proposition 3.3 that $|A(x,\xi )-A(y,\xi )| \leqslant \omega (r)^\theta (|A(y,\xi )|+1) \approx \omega (r)^\theta (\varphi '(y,|\xi |) +1)$ when $|\xi |$ satisfies the condition. $\square$

When considering equations with nonlinearity A it is natural to assume for the function $A^{(-1)}(x,\xi ):=|\xi | A(x,\xi )$. This is the route we took in [42]. However, the next result shows that the conditions for A and $A^{(-1)}$ are equivalent, up to an exponent which does not affect the conclusion of the main results in the next section. Thus we will in the rest of article use the assumptions directly for A.

Proposition 4.9

Let $A:\Omega \times {\mathbb {R}^n} \rightarrow {\mathbb {R}^n}$ have quasi-isotropic (p, q)-growth. Then A satisfies if and only if $A^{(-1)}$ does. If the original modulus continuity is $\omega$, then the new modulus continuity can be taken as $c\,\omega ^{\frac{1}{p'}}$ for some $c=c(p,q,L,L_\omega )>0$.

Proof

Let $x,y\in B_r\cap \Omega$ and $\psi (y,\xi )\in (0,|B_r|^{-1}]$. Assume A satisfies . Since $|\xi | \leqslant c |\xi | |A(y,\xi )|+1$,

$$\begin{aligned}\begin{aligned} |A^{(-1)}(x,\xi )- A^{(-1)}(y,\xi )|&= |\xi | |A(x,\xi )- A(y,\xi )| \\&\leqslant \omega (r) \left( |\xi | |A(y,\xi )|+|\xi |\right) \leqslant c \omega (r) \left( |A^{(-1)}(y,\xi )|+1\right) \end{aligned}\end{aligned}$$

This gives of $A^{(-1)}$, with modulus of continuity $c\omega$.

We prove opposite implication through three cases. If $|\xi |\geqslant 1$, then

$$\begin{aligned} |\xi |\,|A(x,\xi )- A(y,\xi )| \leqslant \omega (r)(|\xi | |A(y,\xi )|+1) \leqslant \omega (r) |\xi | (|A(y,\xi )|+1). \end{aligned}$$

Hence we suppose that $|\xi | < 1$. If further $|A(x,\xi )|,|A(y,\xi )|\leqslant \omega (r)^\frac{1}{p'}$, then

$$\begin{aligned} |A(x,\xi )- A(y,\xi )| \leqslant 2\omega (r)^\frac{1}{p'}. \end{aligned}$$

Otherwise, we use and to deduce $c|\xi |^{p-1} \geqslant \max \{|A(x,\xi )|,|A(y,\xi )|\}\geqslant \omega (r)^\frac{1}{p'}$. Thus $1\leqslant c |\xi |\omega (r)^{-\frac{1}{p}}$ so that

$$\begin{aligned} |\xi |\,|A(x,\xi )- A(y,\xi )| \leqslant \omega (r) (|\xi | |A(y,\xi )|+1) \leqslant c \omega (r)^{1-\frac{1}{p}} |\xi | ( |A(y,\xi )|+1). \end{aligned}$$

Dividing both sides by $|\xi |$ gives the desired estimate. $\square$

Remark 4.10

In [42, Proposition 3.8] we assumed that $A^{(-1)}$ satisfies but in the proof we used the condition for A. This mistake can be corrected by means of the previous proposition.

Having defined the structure conditions, we first consider quasi-isotropic (p, q)-growth for an autonomous function ${\bar{A}}:{\mathbb {R}^n} \rightarrow {\mathbb {R}^n}$ via its trivial extension ${\bar{A}}(x,\xi ):= {\bar{A}}(\xi )$. It has a growth function ${\bar{\varphi }}\in {\Phi _{\textrm{c}}}\cap C^1([0,\infty ))$, cf. [42]. For such ${\bar{A}}$ and ${\bar{\varphi }}$, we present regularity results of weak solutions to

Note that we use the bar-symbol to indicate the autonomous versions of A, F, $\varphi$ and corresponding solutions or minimizers u. In the Uhlenbeck case ${\bar{A}}(\xi )=\frac{{\bar{\varphi }}'(|\xi |)}{\xi }\xi$, we proved the next result in [41] and in [42] we sketched how to extend the proof to the quasi-isotropic case. Since the result is independent of the -type assumptions, it applies directly also to this paper.

Lemma 4.11

($C^{1,\alpha }$-regularity, Lemma 4.4, [42]) Let ${\bar{A}}:\mathbb {R} ^n\rightarrow \mathbb {R} ^n$ have quasi-isotropic (p, q)-growth and ${\bar{\varphi }}\in {\Phi _{\textrm{c}}}$ be its growth function. If ${\bar{u}}{}\in W^{1,{\bar{\varphi }}}(B_r)$ is a weak solution to (${\text {div}}{\bar{A}}$), then $D{\bar{u}}\in C^{0,{\bar{\alpha }}}_{\textrm{loc}}(B_r,{\mathbb {R}^n} )$ for some ${\bar{\alpha }}\in (0,1)$ with the following estimates:

for every $B_\rho \subset B_r$ and $\tau \in (0,1)$. Here ${\bar{\alpha }}$ and $c>0$ depend only on n, p, q and L.

Next, we derive a harmonic approximation lemma for weak solutions and minimizers of autonomous problems. We start by recalling a Lipschitz truncation lemma, which is a formulation from [27, Theorem 3.2] and [8, Theorem 5.2] of the result in [1], see also [19, 25].

Lemma 4.12

For $w\in W^{1,1}_0(B_r)$ and $\lambda >0$ there exist $w_\lambda \in W^{1,\infty }_0(B_r)$ and a zero-measure set N such that $\Vert Dw_\lambda \Vert _{L^\infty (B_r)}\leqslant c \lambda$ for some $c>0$ depending only on n and

$$\begin{aligned} \{w_\lambda \ne w\} \, \subset \, \{ Mw>\lambda \}\cup N, \end{aligned}$$

where M is the Hardy–Littlewood maximal operator, .

Part (1) of the next lemma, on almost solutions, was considered in [27, Lemma 1.1] and [8, Lemma 5.1] in the Orlicz and double phase case, respectively. We streamline and generalize their argument and include also almost minimizers in Part (2). The more precise estimates will allow us to omit the re-scaling step in the proofs of the main theorems.

Lemma 4.13

(Harmonic approximation) Let ${\bar{A}}:{\mathbb {R}^n} \rightarrow {\mathbb {R}^n}$ or ${\bar{F}}:{\mathbb {R}^n} \rightarrow [0,\infty )$ have quasi-isotropic (p, q)-growth, ${\bar{\varphi }}\in {\Phi _{\textrm{c}}}$ be its growth function and ${\bar{u}}\in u+W^{1,{\bar{\varphi }}}_0(B_r)$ be a weak solution to (${\text {div}}{\bar{A}}$), for ${\bar{A}}:=D_\xi {\bar{F}}$ in the case of ${\bar{F}}$. Suppose that there exist $b, \sigma >0$ and $\delta \in (0,1)$ for which

and one of the following conditions holds for all $\eta \in W^{1,\infty }_0(B_r)$:

• (1) for $\mu :=1$.

• (2) for some $\mu >0$.

Then there exist $c=c(n,p,q,L,\sigma )>0$ and $\bar{\sigma }:=\frac{\sigma p}{\mu +\sigma p}$ such that

Proof

All implicit constants in this proof depend only on n, p, q, L and $\sigma$. Let $w:= u-{\bar{u}}\in W^{1,{\bar{\varphi }}}_0(B_r)$ and let $\lambda \geqslant 1$ be a constant to be chosen. With these w and $\lambda$ we consider $w_\lambda \in W^{1,\infty }_0(B_r)$ from Lemma 4.12 so that $\Vert Dw_\lambda \Vert _{L^\infty (B_r)}\lesssim \lambda$. Denote

$$\begin{aligned} {\mathcal {V}}:=\frac{{\bar{\varphi }}'(|Du|+|D{\bar{u}}|)}{|Du|+|D{\bar{u}}|}|Du-D{\bar{u}}|^2,\quad W:=B_r\cap \{w\ne w_\lambda \}\quad \text {and}\quad W^c=B_r\cap \{w=w_\lambda \}. \end{aligned}$$

Since ${\bar{\varphi }}$ is independent of x, we have by a Calderón–Zygmund-type estimate in the Orlicz setting, see for instance [42, Lemma 4.5] with $\theta (x,t)\equiv t^{1+\sigma }$. This and the integrability assumption on Du imply that

To estimate $\frac{|W|}{|B_r|}$, we use $W\subset \{M(|Dw|)>\lambda \}\cup N$ from Lemma 4.12 and the maximal estimate in $L^{{\bar{\varphi }}^{1+\sigma }}$ [33, Corollary 4.3.3]:

where, in the last step we used ${\bar{\varphi }}(|Dw|)\lesssim {\bar{\varphi }}(|D{\bar{u}}|)+{\bar{\varphi }}(|Du|)$ and the earlier integrablilty estimates for Du and $D{\bar{u}}$. Using Hölder’s inequality and these estimates, we find that

(4.14)

We first assume (1). Using monotonicity (4.6) and that ${\bar{u}}$ is a weak solution to (${\text {div}}{\bar{A}}$), we find that

For the first term we use assumption (1) with $\eta =w_\lambda$, and for the second we use that growth functions satisfy $|{\bar{A}}|\lesssim {\bar{\varphi }}'$. Continuing using Young’s inequality, ${\bar{\varphi }}^*({\bar{\varphi }}'(t))\lesssim {\bar{\varphi }}(t)$ and $\Vert Dw_\lambda \Vert _{L^\infty (B_r)}\lesssim \lambda$, we find that

In W, we use ${\mathcal {V}}\lesssim {\bar{\varphi }}(|Du|)+{\bar{\varphi }}(|D{\bar{u}}|)$ and obtain the same integral as on the right-hand side, above. With these estimates and (4.14), we obtain that

We choose $\lambda :={\bar{\varphi }}^{-1}(b\delta ^{-\kappa })\leqslant \delta ^{-\kappa /p}{\bar{\varphi }}^{-1}(b)$ with $\kappa :=\frac{p}{1+\sigma p}$, and find that

This is the desired upper bound with $\bar{\sigma }:= \frac{\sigma p}{1+\sigma p}$ and concludes the proof in Case (1).

Assume next that ${\bar{A}}:=D_\xi {\bar{F}}$ and (2) holds. We showed in [42, Lemma 6.3] that

$$\begin{aligned} \frac{{\bar{\varphi }}'(|\xi _1|+|\xi _2|)}{|\xi _1|+|\xi _2|}|\xi _1-\xi _2|^2 \lesssim {\bar{F}}(\xi _1)-{\bar{F}}(\xi _2) -D_\xi {\bar{F}}(\xi _2) \cdot (\xi _1-\xi _2). \end{aligned}$$

Using this with ${\bar{A}}=D_\xi {\bar{F}}$, the weak form of (${\text {div}}{\bar{A}}$), ${\bar{u}}=u+w_\lambda$ in $W^c$, ${\bar{F}}\approx {\bar{\varphi }}$, assumption (2), $|{\bar{A}}|\lesssim {\bar{\varphi }}'$ and Young’s inequality with ${\bar{\varphi }}^*({\bar{\varphi }}'(t))\lesssim {\bar{\varphi }}(t)$, we have

In W we use the estimate ${\mathcal {V}}\lesssim {\bar{\varphi }}(|Du|)+{\bar{\varphi }}(|D{\bar{u}}|)$ as before. With (4.14) and the choice $\lambda :={\bar{\varphi }}^{-1}(b\delta ^{-\kappa })\lesssim \delta ^{-\kappa /p} {\bar{\varphi }}^{-1}(b)$ for $\kappa :=\frac{p}{\mu +\sigma p}$, we obtain that

This is the desired upper bound with ${\bar{\sigma }}:= \frac{\sigma p}{\mu +\sigma p}$ and concludes the proof in Case (2). $\square$

5 Maximal regularity

Let $B_r\Subset \Omega$ with $|B_r|\leqslant 1$, $\gamma \in (0,1)$ and

$$\begin{aligned} t_K:= K |B_r|^{\frac{\gamma -1}{n}}\quad \text {for fixed }\ K\geqslant 1. \end{aligned}$$

For $\varphi \in {\Phi _{\textrm{c}}}(\Omega )$ with $\varphi '$ satisfying , and with $1<p\leqslant q_1$ we define

$$\begin{aligned} {\bar{\varphi }}(t):=\int _0^t {\bar{\varphi }}'(s)\,ds \quad \text {with}\quad {\bar{\varphi }}'(t):= {\left\{ \begin{array}{ll} \varphi '(x_0,t)&{}\text {if}\ \ 0\leqslant t\leqslant t_K,\\ \frac{\varphi '(x_0,t_K)}{t_K^{p-1}}t^{p-1}&{}\text {if} \ \ t_K\leqslant t, \end{array}\right. } \end{aligned}$$

(5.1)

where $x_0$ is the center of $B_r=B_r(x_0)$. The relationship between $\varphi$ and ${\bar{\varphi }}$ is analogous to that in [41, Section 5] with $t_1=0$ and $t_2=t_K$. Due to our improved tools, we are able to simplify the argument concerning small values of t by removing the case $t\in [0,t_1]$.

Proposition 5.2

Let $t_K$, $\varphi$ and ${\bar{\varphi }}$ be as above. Suppose that $\varphi ^+_{B_r}(t)\leqslant L_K \varphi ^-_{B_r}(t)$ for some $L_K\geqslant 0$ and all $t \in [1, t_K]$.

• (1) ${\bar{\varphi }}\in C^1([0,\infty ))$ and ${\bar{\varphi }}'$ satisfies and .

• (2) ${\bar{\varphi }}(t)= \varphi (x_0,t)$ for all $t\leqslant t_K$.

• (3) ${\bar{\varphi }}(t) \leqslant \frac{q_1}{p} L_K\varphi (x,t)+L$ for all $(x,t)\in B_r\times [0,\infty )$ and $W^{1,\varphi }(B_r) \subset W^{1,{\bar{\varphi }}}(B_r)$.

Proof

Parts (1) and (2) follow directly from the definition of ${\bar{\varphi }}$ and the inclusion in (3) follows from the inequality. If $t\leqslant 1$, then the inequality in (3) follows from and when $t\in [1,t_K]$, it follows from $\varphi ^+_{B_r}(t)\leqslant L_K \varphi ^-_{B_r}(t)$ and (2). If $t > t_K$, we calculate

$$\begin{aligned} \begin{aligned} {\bar{\varphi }}(t)&= \varphi (x_0,t_K) + \int _{t_K}^t \frac{\varphi '(x_0,t_K)}{t_K^{p-1}}s^{p-1}\, ds = \varphi (x_0,t_K) + \frac{t^p-t_K^p}{ t_K^p} \frac{t_K \varphi '(x_0,t_K)}{p}\\&\leqslant \frac{q_1}{p}\Big (\frac{t}{t_K}\Big )^p \varphi (x_0,t_K) \leqslant \frac{q_1}{p}L_K\Big (\frac{t}{t_K}\Big )^p \varphi ^-_{B_r}(t_K) \leqslant \frac{q_1}{p}L_K \varphi (x,t). \square \end{aligned} \end{aligned}$$

We prove our main theorem on Hölder continuous weak solutions to (${\text {div}}A$). The major novelties as follows: We use Proposition 3.3 to deal with large values of the derivative; this allows us to handle the borderline double phase case $q-p=\frac{\alpha }{1-\gamma }$ with $a\in VC^{0,\alpha }$. Second, this is the first time that harmonic approximation from [8] has been applied in the generalized Orlicz case. Third, optimizations in the formulations allow us to avoid many steps in previous proofs and present a much more streamlined argument. Fourth, we obtain $C^{1,\alpha }$-regularity with exponent $\alpha$ independent of the a priori information $[u]_\gamma$.

Theorem 5.3

Let $A:\Omega \times {\mathbb {R}^n} \rightarrow {\mathbb {R}^n}$ have quasi-isotropic (p, q)-growth and $u\in W^{1,1}_{\textrm{loc}}(\Omega )\cap C^{0,\gamma }(\Omega )$ be a weak solution to (${\text {div}}A$) with $\gamma \in (0,1)$.

• (1) If A satisfies , then $u\in C^{0,\alpha }_{\textrm{loc}}(\Omega )$ for every $\alpha \in (0,1)$.

• (2) If A satisfies with $\omega (r)\lesssim r^{\beta }$ for some $\beta >0$, then $u\in C^{1,\alpha }_{\textrm{loc}}(\Omega )$ for some $\alpha =\alpha (n,p,q,L,\gamma ,\beta )\in (0,1)$.

Proof

We prove the result in three steps. All implicit constants in the following estimates depend only on n, p, q, L and $[u]_\gamma$.

Step 1, setting and approximating equation. For $\omega$ from we fix $B_{4r}\Subset \Omega$ with

$$\begin{aligned} r \in (0, 1),\quad \omega (4r)\leqslant 1 \quad \text {and}\quad \omega (r)^{1-\theta }\leqslant \tfrac{1}{3}, \end{aligned}$$

(5.4)

where $\theta \in (0,1)$ is given in Step 2, below. Since we always consider $B_r$ with $\omega (4r)\leqslant 1$, we assume without loss of generality that A satisfies with $L_\omega =1$. We set

$$\begin{aligned} K:=3^{-(1-\gamma )} \omega (r)^{-(1-\theta )(1-\gamma )} \quad \text {so that}\quad t_K=3^{-(1-\gamma )} \omega (r)^{-(1-\theta )(1-\gamma )} |B_r|^{\frac{\gamma -1}{n}}. \end{aligned}$$

Let $\varphi \in {\Phi _{\textrm{c}}}(\Omega )$ be the growth function of A from Proposition 4.5 so that $\varphi (x,\cdot )\in C^1([0,\infty ))$ and $\varphi '$ satisfies , and . By Proposition 4.8(1) &(3), $\varphi$ satisfies with $L_\omega$ depending only on p, q and L and

$$\begin{aligned} |A(x,\xi )-A(y,\xi )|\leqslant \omega (r)^\theta \big (\varphi '(y,|\xi |)+1\big ) \quad \text {when }\ |\xi | \leqslant t_K. \end{aligned}$$

By of $\varphi '$, $|A|\approx \varphi '$ and $\omega (r)\leqslant 1$ we conclude that

$$\begin{aligned} \varphi (x,t) \approx t |A(x,te_1)| \leqslant t (|A(y,te_1)|+\varphi '(y,t)+1) \lesssim \varphi (y,t) \end{aligned}$$

for every $t\in [1, t_K]$ and $x,y\in B_r$. Thus we can apply Proposition 5.2 with $L_K>0$ depending only on p, q and L.

Let ${\bar{\varphi }}\in {\Phi _{\textrm{c}}}$ be from (5.1). By [42, Lemma 5.2] with $t_1=0$ and $t_2=t_K$ there exists an autonomous nonlinearity ${\bar{A}}\in C({\mathbb {R}^n} ,{\mathbb {R}^n} )\cap C^{1}({\mathbb {R}^n} {\setminus }\{ 0\},{\mathbb {R}^n} )$ having quasi-isotropic $(p,q_1)$-growth such that ${\bar{\varphi }}$ is its growth function and

$$\begin{aligned} {\bar{A}}(\xi )=A(x_0,\xi ) \quad \text {whenever }|\xi | \leqslant \tfrac{1}{2}t_K. \end{aligned}$$

The exact form of ${\bar{A}}$ is given in [42, (5.2)].

By Theorem 3.14, $\varphi (\cdot ,|Du|)\in L^{1+\sigma }(B_r)$ for some $\sigma >0$ depending only on n, p, q, L, and by (3.13) in $B_{2r}$,

(5.5)

Let ${\bar{u}}\in W^{1,{\bar{\varphi }}}(B_r)$ be the weak solution to (${\text {div}}{\bar{A}}$) in $B_r$ with boundary value u. Then ${\bar{u}}$ is a quasiminimizer of the ${\bar{\varphi }}$-energy and so the results of Sect. 3 can be applied. By Jensen’s inequality, the minimization property and Proposition 5.2(3),

Then we use the reverse Hölder inequality (Theorem 3.14), , Proposition 5.2(2) with (5.5) and of $\varphi$ to conclude that

It follows that

(5.6)

Next we set

By the reverse Hölder inequality (Theorem 3.14) and the earlier estimate (5.6),

(5.7)

and, since $J\gtrsim 1$,

(5.8)

Step 2, harmonic approximation. We prove that u is an almost weak solution to (${\text {div}}{\bar{A}}$) in the sense that

for some $c\geqslant 1$ depending on n, p, q, L and $[u]_{\gamma ,4r}$ and all $\eta \in W^{1,\infty }_0(B_r)$.

It suffices to consider $\eta$ with $\Vert D\eta \Vert _\infty \leqslant 1$ by scaling. Since u is a weak solution to (${\text {div}}A$),

where $\displaystyle E_1:=\left\{ x\in B_r: |Du| \leqslant \tfrac{1}{2} t_K\right\}$ and $\displaystyle E_2:= \left\{ x\in B_r: |Du|> \tfrac{1}{2} t_K\right\}$.

We first consider $E_1$ so that ${\bar{A}}=A(x_0,\cdot )$. By Proposition 4.8(3) with $s:=\frac{n}{1-\gamma }$,

We abbreviate ${\hat{\varphi }}:=\varphi ^+_{B_{r}}$ and estimate the integral on the right-hand side by $\varphi '(x,t)\approx \varphi (x,t)/t$, Jensen’s inequality for ${\hat{\varphi }}^*$, ${\hat{\varphi }}^*( \varphi (x,t)/t) \leqslant \varphi (x,t)$, Hölder’s inequality and estimate (5.8) for J:

Since $1\lesssim \tfrac{{\bar{\varphi }}(J)}{J}$, the whole integral over $E_1$ can be bounded by $\omega (r)^\theta \tfrac{{\bar{\varphi }}(J)}{J}$.

In $E_2$ we estimate

Since $\frac{{\bar{\varphi }}(J)}{{\bar{\varphi }}(\frac{1}{2} t_K)} \lesssim \frac{{\bar{\varphi }}(r^{\gamma -1})}{{\bar{\varphi }}( Kr^{\gamma -1})} \leqslant K^{-p}$ by (5.7) and , we obtain

(5.9)

By Proposition 5.2(3) and , ${\bar{\varphi }}(t)\lesssim \varphi (x,t)$ when $t\geqslant 1$. Using this, the ${\hat{\varphi }}^*$-Jensen inequality as in $E_1$, the previous estimate and of $({\hat{\varphi }}^*)^{-1}$, we find that

where we used $({\hat{\varphi }}^*)^{-1}({\hat{\varphi }}(t))\approx {\hat{\varphi }}(t)/t$ and (5.8) in the last step. The desired estimate follows when we combine the estimates in $E_1$ and $E_2$, recall that $K\geqslant 3^{-1} \omega (r)^{-(1-\theta )(1-\gamma )}$ and choose $\theta :=\frac{(1-\gamma )\sigma (p-1)}{(1-\gamma )\sigma (p-1)+1}$.

Step 3, conclusion. Applying Lemma 4.13(1) with $b:={\bar{\varphi }}(J)$ to the inequality from Step 2, we obtain that

where $\bar{\sigma }=\frac{\sigma p}{1+\sigma p}$. By well-known techniques (see, e.g., [41, Corollary 6.3] for details) this implies an $L^1$-estimate for the difference of the gradients of u and ${\bar{u}}$ from Step 1:

(5.10)

with $c\geqslant 1$ depending on n, p, q, L and $[u]_{\gamma , 4r}$ and $\bar{\omega }$ from Step 2. Note that we can make $\bar{\omega }(r)^{\frac{\bar{\sigma }}{2q_1}}$ as small as we want by choosing r small. Therefore, this inequality and the Lipschitz regularity of the ${\bar{A}}$-solutions ${\bar{u}}$ (the first estimate in Lemma 4.11) with (5.6) imply local $C^{0,\alpha }$-regularity for every $\alpha \in (0,1)$ by known methods, see, e.g., [41, Theorem 7.2].

We next assume that $\omega (r)\lesssim r^{\beta }$ for some $\beta >0$. Fix $\Omega '\Subset \Omega$. From Part (1) we obtain $u\in C^{0, \gamma '}(\Omega ')$ for any $\gamma '\in (\gamma ,1)$, and consider $B_{4r}\subset \Omega '$ with r satisfying (5.4) and $[u]_{\gamma ,4r,\Omega '}\leqslant 1$. Then we obtain (5.10) with $\bar{\omega }(r)^{\frac{\bar{\sigma }}{2q_1}}\lesssim r^{\beta _0}$ for $\beta _0$ depending only n, p, q, L, $\gamma$ and $\beta$. This inequality and the Hölder regularity of the gradient of the ${\bar{A}}$-solution ${\bar{u}}$ (the second estimate in Lemma 4.11) with (5.6) imply $u\in C^{1,\alpha }_{\textrm{loc}}(\Omega ')$ for any $\alpha \in (0, \min \{\beta _0,1-\gamma '\})$ by known methods, see, e.g., [41, Theorem 7.4]. Since $\beta _0$ and $1-\gamma '$ are independent of the arbitrary set $\Omega '\Subset \Omega$, this implies that $u\in C^{1,\alpha }_{\textrm{loc}}(\Omega )$ for some $\alpha \in (0,1)$ depending only on n, p, q, L, $\gamma$ and $\beta$. $\square$

Next we prove maximal regularity for Hölder continuous minimizers of ($\min F$). This is our second main result.

Theorem 5.11

Let $F:\Omega \times {\mathbb {R}^n} \rightarrow \mathbb {R}$ have quasi-isotropic (p, q)-growth and $u\in W^{1,1}_{\textrm{loc}}(\Omega )\cap C^{0,\gamma }(\Omega )$ be a minimizer of ($\min F$) with $\gamma \in (0,1)$.

• (1) If F satisfies , then $u\in C^{0,\alpha }_{\textrm{loc}}(\Omega )$ for every $\alpha \in (0,1)$.

• (2) If F satisfies with $\omega (r)\lesssim r^{\beta }$ for some $\beta >0$, then $u\in C^{1,\alpha }_{\textrm{loc}}(\Omega )$ for some $\alpha =\alpha (n,p,q,L,\gamma ,\beta )\in (0,1)$.

Proof

The methodology is similar to Theorem 5.3 except for the application of harmonic approximation. Hence we will take advantage many parts of that proof.

Step 1, setting and approximating functional. We use the same choice of r and K as in Theorem 5.3 and define J in the same way. Let $\varphi \in \Phi _c(\Omega )$ be the growth function of F from Proposition 4.5; it satisfies the same properties as in Theorem 5.3. In [42, Lemma 5.3] we constructed an autonomous function ${\bar{F}}:{\mathbb {R}^n} \rightarrow [0,\infty )$ such that ${\bar{\varphi }}\in {\Phi _{\textrm{c}}}$ from (5.1) is its growth function and

$$\begin{aligned} {\bar{F}}(\xi )=F(x_0,\xi ) \quad \text {whenever } |\xi | \leqslant \tfrac{1}{2} t_K. \end{aligned}$$

By Theorem 3.14, $\varphi (\cdot ,|Du|)\in L^{1+\sigma }(B_r)$ for some $\sigma =\sigma (n,p,q,L,[u]_{\gamma ,4r})\in (0,1)$. Let ${\bar{u}}\in W^{1,{\bar{\varphi }}}(B_r)$ be the minimizer of

$$\begin{aligned} \min _{{\bar{u}}\in u+W^{1,{\bar{\varphi }}}_0(B_r)} \int _{B_r} {\bar{F}}(D{\bar{u}})\,dx, \end{aligned}$$

(5.12)

or, equivalently, the weak solution to (${\text {div}}{\bar{A}}$) in $B_r$ with ${\bar{A}}:=D_\xi {\bar{F}}$ and boundary value given by u.

Step 2, harmonic approximation. We prove that u is an almost minimizer of (5.12) in the sense that there exists $c=c(n,p,q,L,[u]_{\gamma ,4r})\geqslant 1$ such that

for every $\eta \in W^{1,\infty }_0(B_r)$, where $\bar{\omega }(r):= \omega (r)^{\frac{(1-\gamma )\sigma p}{(1-\gamma )\sigma p+1}}$.

Let $E_1$ and $E_2$ be the sets from Step 2 of the proof of Theorem 5.3. By ${\bar{F}}\approx {\bar{\varphi }}$ (4.7), the definition of ${\bar{F}}$, Propositions 5.2(3) and 4.8(3), and the definition of J,

where the estimate for the term in $E_2$ is from (5.9).

Next we obtain a similar estimate for $v:=u+\eta \in u + W^{1,\infty }_0(B_r)$. We define sets $E_i'$ like $E_i$ but with |Du| replaced by |Dv|. Since u is an F-minimizer and $F\approx \varphi$ (4.7),

In $E_1'$ we use Proposition 4.8(3) with ${\bar{F}}=F(x_0,\cdot )$:

For the second term on the right-hand side, we use $v=u+\eta$ and $\varphi (x,|Dv|)\lesssim \varphi (x,|Du|)+\varphi (x,|D\eta |)$. Thus

We use of $\varphi$ along with (5.8) to handle the integral with $D\eta$:

In $E_2'$ we estimate

With the estimate for $D\eta$ from the previous paragraph, this and $\frac{{\bar{\varphi }}(J)}{{\bar{\varphi }}(\frac{1}{2} t_K)} \lesssim K^{-p}$ give

Collecting the estimates from this and the previous paragraph, we arrive at

Combining this with the previous paragraphs, the estimate $K\geqslant 3^{-1}\omega (r)^{-(1-\theta )(1-\gamma )}$ and the choice of $\theta :=\frac{(1-\gamma )\sigma p}{(1-\gamma )\sigma p+1}$, we complete this step.

Step 3, conclusion. We apply Lemma 4.13(2) with $b:={\bar{\varphi }}(J)$ and $\mu :=(1+\sigma )q_1$ to conclude that

where $\bar{\sigma }:=\frac{\sigma p}{(1+\sigma )q_1 +\sigma p}$ and $\bar{\omega }$ is from Step 2. The estimate

follows from this as in [41, Corollary 6.3]. We complete the proof as in Step 3 of the proof of Theorem 5.3. Hence we omit details. $\square$

We apply the previous theorem to the double phase energies from Example 3.2.

Corollary 5.13

Let $1<p\leqslant q$, $a\geqslant 0$ and $\gamma \in (0,1)$. Assume that either

• $F(x,\xi ):=|\xi |^p + a(x) |\xi |^q$, where $a\in VC^{0,\alpha }(\Omega )$, $\alpha \in (0,1)$ and $q-p \leqslant \frac{1}{1-\gamma }\alpha$; or

• $F(x,\xi ):= |\xi |^p + a(x)^\alpha |\xi |^q$, where $a\in C^{0,1}(\Omega )$, $\alpha >1$ and $q-p < \frac{1}{1-\gamma }\alpha$.

If $u\in W^{1,1}_{\textrm{loc}}(\Omega )\cap C^{0,\gamma }(\Omega )$ is a minimizer of ($\min F$), then $u\in C^{1,\beta }_{\textrm{loc}}(\Omega )$ for some $\beta$ depending only on n, p, q, $\alpha$ and $\gamma$.

Proof

Consider first the case with $\alpha <1$. By Example 3.2, F satisfies . Therefore $u\in C^{0,\beta }_{\textrm{loc}}(\Omega )$ for every $\beta \in (0,1)$ by the previous theorem. Fix $\gamma _1\in (\gamma ,1)$. Then F satisfies with $\omega (r)\lesssim r^{\alpha -(q-p)(1-\gamma _1)}$ and $\alpha -(q-p)(1-\gamma _1)> \alpha -(q-p)(1-\gamma )\geqslant 0$. Since $u\in C^{0,\gamma _1}_\textrm{loc}(\Omega )$, the previous theorem yields that $u\in C^{1,\beta }_{\textrm{loc}}(\Omega )$ for some $\beta \in (0,1)$ depending on n, p, q, L, $\alpha$ and $\gamma$. In the case $\alpha >1$ we can directly apply the theorem since the strict inequality $q-p < \frac{1}{1-\gamma }\alpha$ implies that the condition holds with a $\omega$ of power-type (cf. Example 3.2). $\square$

Finally, combining Theorems 3.12, 5.3 and 5.11, we obtain regularity results for BMO and $L^{s^*}$ weak solutions and minimizers. Note that the case $L^{s^*}$ with $s\leqslant (n(1-\frac{p}{q})$ remains open, due to a lack of a Sobolev–Poincaré inequality.

Corollary 5.14

Let $A:\Omega \times {\mathbb {R}^n} \rightarrow {\mathbb {R}^n}$ or $F:\Omega \times {\mathbb {R}^n} \rightarrow \mathbb {R}$ have quasi-isotropic (p, q)-growth and satisfy . Suppose that $u\in W^{1,1}_{\textrm{loc}}(\Omega )$ is a weak solution to (${\text {div}}A$) or a minimizer of ($\min F$) and that one of the following holds:

• (1) $s=n$ and $u\in BMO(\Omega )$,

• (2) $s\in (n(1-\frac{p}{q}), n)$ and $u\in L^{s^*}(\Omega )$.

Then $u\in C^{0,\alpha }_{\textrm{loc}}(\Omega )$ for every $\alpha \in (0,1)$.

Furthermore, if holds with $\omega (r)\lesssim r^{\beta }$ for some $s'>s$ and $\beta >0$, then $u\in C^{1,\alpha }_{\textrm{loc}}(\Omega )$ for some $\alpha \in (0,1)$ depending on n, p, q, L, s, $s'$ and $\beta$.

Proof

Note that implies and fix $\Omega '\Subset \Omega$. Since u is a quasiminimizer of the isotropic problem, Theorem 3.12 implies that $u\in C^{0,\gamma }(\Omega ')$ for some $\gamma \in (0,1)$. The condition with $s<\frac{n}{1-\gamma }$ implies . By Theorem 5.3 or 5.11, we obtain $u\in C^{0,\alpha }_{\textrm{loc}}(\Omega ')$ so that $u\in C^{0,\alpha }_{\textrm{loc}}(\Omega )$ for every $\alpha \in (0,1)$.

Next, we suppose that $\omega (r)\lesssim r^{\beta }$ in and fix $\Omega '\Subset \Omega$. By the previous part, $u\in C^{0,\gamma }(\Omega ')$ for any $\gamma \in (0,1)$. Furthermore, holds with $\omega (r)\lesssim r^{\beta }$ when $\gamma$ is chosen so large that $\frac{n}{1-\gamma }\geqslant s'$. Therefore, by Theorem 5.3 or 5.11, we obtain $u\in C^{1,\alpha }_{\textrm{loc}}(\Omega ')$ so that $u\in C^{1,\alpha }_{\textrm{loc}}(\Omega )$ for some $\alpha \in (0,1)$ depending only on n, p, q, L, $\gamma$ and $\beta$. $\square$

Remark 5.15

For $C^{1,\alpha }$-regularity, the obtained exponent $\alpha$ is independent of the a priori information on weak solutions or minimizers.