Regularity theory for non-autonomous partial differential equations without Uhlenbeck structure

We establish maximal local regularity results of weak solutions or local minimizers of \[ \operatorname{div} A(x, Du)=0 \quad\text{and}\quad \min_u \int_\Omega F(x,Du)\,dx, \] providing new ellipticity and continuity assumptions on $A$ or $F$ with general $(p,q)$-growth. Optimal regularity theory for the above non-autonomous problems is a long-standing issue; the classical approach by Giaquinta and Giusti involves assuming that the nonlinearity $F$ satisfies a structure condition. This means that the growth and ellipticity conditions depend on a given special function, such as $t^p$, $\varphi(t)$, $t^{p(x)}$, $t^p+a(x)t^q$, and not only $F$ but also the given function is assumed to satisfy suitable continuity conditions. Hence these regularity conditions depend on given special functions. In this paper we study the problem without recourse to special function structure and without assuming Uhlenbeck structure. We introduce a new ellipticity condition using $A$ or $F$ only, which entails that the function is quasi-isotropic, i.e.\ it may depend on the direction, but only up to a multiplicative constant. Moreover, we formulate the continuity condition on $A$ or $F$ without specific structure and without direct restriction on the ratio $\frac qp$ of the parameters from the $(p,q)$-growth condition. We establish local $C^{1,\alpha}$-regularity for some $\alpha\in(0,1)$ and $C^{\alpha}$-regularity for any $\alpha\in(0,1)$ of weak solutions and local minimizers. Previously known, essentially optimal, regularity results are included as special cases.


Introduction
Research on regularity of weak solutions or minimizers of the problems divA(x, Du) = 0 and min uˆΩ F (x, Du) dx is a major topic in the partial differential equations and the calculus of variations. If there is no direct dependence on x (i.e., A(x, ξ) ≡ A(ξ) and F (x, ξ) ≡ F (ξ)), these are called autonomous problems. The simplest non-linear model cases is the p-power function F (ξ) = |ξ| p , 1 < p < ∞, and the corresponding Euler-Lagrange equation is the p-Laplace equation where A(Du) = |Du| p−2 Du. The maximal regularity of weak solutions of the p-Laplace equation is C 1,α for some α ∈ (0, 1) depending only on p and the dimension n (e.g., [14,17,31,39]).
For non-autonomous problems, there is also direct x-dependence. To tackle this case, Giaquinta and Giusti [20,21] introduced the following p-type growth conditions: Here, F is related to the perturbed case a(x)|ξ| p and has the same p-type growth at all points. Lieberman [32] generalized the model by replacing |ξ| p with Orlicz growth ϕ(|ξ|). Marcellini [34] introduced non-standard, so-called (p, q)-growth where the exponent p on the right-hand side is replaced by q > p. In this situation, we need to assume that q p is close to 1, see, e.g., [3,4,12,35]. However, all these structure conditions fail to accommodate many kinds of energy functionals since the variability in the x-and ξ-direction are treated separately.
In [28], we introduced a different approach which does not impose any direct restriction on q p . However, we were only able to prove maximal regularity for local minimizers when F (x, ξ) = F (x, |ξ|) has so-called Uhlenbeck structure. In this article we extend the results to both minimizers and weak solutions and dispense with the Uhlenbeck restriction.
We collect some conditions for A : Ω × R n → R n and F : Ω × R n → R with an open set Ω ⊂ R n (n 2), which determine our equation and minimization energy, respectively. See Section 2 for further definitions and notation, including the continuity assumption (wVA1) that is the other main assumption. Assumption 1.2. We say that A : Ω × R n → R n satisfies Assumption 1.2 if the following three conditions hold: (1) For every x ∈ Ω, A(x, 0) = 0, A(x, ·) ∈ C 1 (R n \ {0}, R n ) and for every ξ ∈ R n , A(·, ξ) is measurable.
(2) There exist L 1 and 1 < p < q such that the radial function t → |D ξ A(x, te)| satisfies (A0), (aInc) p−2 and (aDec) q−2 with the constant L, for every x ∈ Ω and e ∈ R n with |e| = 1. (The (p, q)-growth condition) (3) There exists L 1 such that |D ξ A(x, ξ ′ )| L D ξ A(x, ξ)e · e for all x ∈ Ω and ξ, ξ ′ , e ∈ R n with |ξ| = |ξ ′ | = 0 and |e| = 1. (The quasi-isotropic ellipticity condition) The (A0) condition in (2) means that a coefficient factor of A is nondegenerate and nonsingular, for instance a ≈ 1 when A(x, ξ) = a(x)|ξ| p−2 ξ. The (aDec) q−2 and (aInc) p−2 conditions in (2) are equivalent to the function t → t 2 |D ξ A(x, te)| satisfying the ∆ 2 -and ∇ 2 -conditions, respectively. In particular, we note from (2) that D ξ A(x, ξ) = 0 when ξ = 0. Uhlenbeck structure has been replaced by (3), which is a quasi-isotropy condition since different directions behave the same up to a constant. It is known that completely anisotropic equations do not necessarily have any regularity as solutions may even be locally unbounded [19]. We also note that if A(x, ξ) = D ξ F (x, ξ) the condition (3) means that the Hessian matrix D 2 ξ F (x, ξ) with ξ = 0 is positive definite and all its eigenvalues on each sphere for ξ are comparable uniformly in x and the radii of spheres, that is, 1 sup{eigenvalues of D 2 ξ F (x, te) : |e| = 1} inf{eigenvalues of D 2 ξ F (x, te) : |e| = 1} L for each x ∈ Ω and t > 0, whereL depends only on L and n. Compare this to the p-growth condition in (1.1), which implies a stronger condition, where x is inside the supremum and infimum: L for each t > 0.
We further refer to [12,13] for a discussion of non-uniformly elliptic conditions in terms of ratios of eigenvalues (in particular [12,Section 4.6] and [13,Section 1]) and related regularity results (see also Remark 1.8).
Remark 1.4. The continuity condition (wVA1) for Φ-functions was introduced in [28], see also Section 2.2. In our knowledge, the above theorem covers all previous known C 1,α -regularity results for equation (divA) with α independent of the solution. Some examples of functionals satisfying Assumptions 1.2 or 1.6 are presented in Section 7.
3 Remark 1.5. In (wVA1), ε > 0 is arbitrary and it is possible that β ε → 0 as ε → 0. Moreover, the constantL =L K is from (wVA1) and K > 0 is arbitrary. However, for given structure we actually choose certain, positive ε and K (see Section 6, in particular, (6.1) and (6.2)), and α in the previous theorem depends on this β ε andL =L K . The same applies in Theorem 1.7.
If equation (divA) is an Euler-Lagrange equation, that is, if A = D ξ F for some a function F : Ω × R n → R, then we can consider milder regularity assumptions in the context of variational calculus. Assumption 1.6. We say that F : Ω×R n → [0, ∞) satisfies Assumption 1.6 if the following two conditions hold: (2) and (3) of Assumption 1.2.
Under our differentiability assumptions on F , u ∈ W 1,1 loc (Ω) is a local weak solution to (divA) with A = D ξ F if and only if it is a local minimizer of (min F ). Since not every mapping A is of the form D ξ F , Theorem 1.3 is more general in terms of structure. On the other hand, (wVA1) with G(x, ξ) = |ξ|D ξ F (x, ξ) implies (wVA1) with G(x, ξ) = F (x, ξ), but not the other way around (cf. Proposition 3.15), so in that sense the assumption of Theorem 1.7 is weaker. Remark 1.8. De Filippis and Mingione [13] study Hölder regularity of the gradient of minimizers of non-autonomous, (p, q)-growth functionals with an upper bound of q p . Their condition is not covered by the condition in the above theorem, but their Hölder exponent may depend on minimizers.
Let us briefly outline the rest of the paper and point out the main novelties. We first collect some background information in next section. In Section 2.2, we introduce our new conditions including (wVA1) that have been adapted to the non-Uhlenbeck case. Formulating appropriate conditions and unifying them to cover all the cases was the first challenge that we faced. In Section 3, we construct for A or F an approximating function ϕ with Uhlenbeck structure (e.g., F (x, ξ) ≈ ϕ(x, |ξ|) in the functional case) that we call a growth function. The function ϕ is similar to the one used in [28] which allows us to use some earlier results. However, ϕ does not have the same continuity property (wVA1) as A or F .
In Section 4, we consider regularity results in two simpler cases that are used as building blocks later on. Specifically, we show that our weak solution or minimizer is also a quasiminimizer of a non-autonomous functional with Uhlenbeck structure and we study related A-equations orF -energy functionals which are autonomous but lack Uhlenbeck structure. The main difficulty for us was constructing an appropriate approximating autonomous problem with autonomous functionsĀ orF from A or F and a growth functionφ. This is solved in Section 5. For the function ϕ, we use the same approximationφ as in [28]. However, for A and F we need a novel approach of transitioning smoothly between different growth using appropriately chosen functions η i . With these elements in place, we prove the main results in Section 6 using comparison and iteration arguments. In the final section, we apply the main result in the special cases of variable exponent-and double phase-type growth.

Preliminaries
2.1. Notation and definitions. Throughout the paper we always assume that Ω is a bounded domain in R n . Let x 0 ∈ R n and r > 0. Then B r (x 0 ) is the standard open ball in R n centered at x 0 with radius r. If its center is clear, we simply write respectively, for all 0 < t < s < ∞. In particular, if L = 1, we say f is increasing or decreasing. For an integrable function f in U ⊂ R n , we define ffl U f dx := 1 |U |´U f dx as the average of f in U in the integral sense. For functions f, g : U → R, f g or f ≈ g (in U) mean that there exists C 1 such that f (y) Cg(y) or C −1 f (y) g(y) Cf (y), respectively, for all y ∈ U. By D = D x we denote the derivative with respect to the space-variable x. The same idea and notation is also used for F : Ω × R N → [0, ∞]. If the map t → ϕ(x, t), t 0, is increasing for every x ∈ Ω, then the (left-continuous) inverse function with respect to t is defined by ϕ −1 (x, t) := inf{τ 0 : ϕ(x, τ ) t}. If ϕ is strictly increasing and continuous in t, then this is just the normal inverse function.
Definition 2.1. We define some conditions for ϕ : Ω × [0, ∞) → [0, ∞) and γ ∈ R related to regularity with respect to the second variable, which are supposed to hold for all x ∈ Ω and a constant L 1 independent of x.
We next introduce classes of Φ-functions and generalized Orlicz spaces. We refer to [22] for more details about the basics. In the sequel we omit the words "generalized" and "weak" mentioned in parentheses.
We note that convexity implies (Inc) 1 hence Φ c (Ω) ⊂ Φ w (Ω). While we are mainly interested in convex functions for minimization problems and related PDEs, the class Φ w (Ω) is very useful for constructing approximating functionals, as it allows much more flexibility. This will be utilized several times in this article.
We generalize the conditions to the non-Uhlenbeck situation. It can be easily seen that (VA1)=⇒(wVA1)=⇒(A1). The results of this paper require only (wVA1), but (VA1) is included since it is simpler to check and is a sufficient condition.
for all x, y ∈ B r ∩ Ω and ξ ∈ R n . We say that G satisfies: Here, a modulus of continuity ω means that ω is concave and nondecreasing and satisfies If we compare these conditions in the case N = 1 with the previously mentioned conditions in earlier papers, we see that they do not look exactly the same. First of all the earlier conditions correspond to the case K = 1 only. If we consider quasi-convex domains Ω and ϕ satisfies (aDec), then the condition with any K is equivalent to the condition with fixed K. This is proved by a simple chain argument, see [23,Lemma 3.3] for (A1).
Furthermore, the earlier conditions did not allow ξ satisfying |G(y, ξ)| < ω(r). For instance, the old formulation of (VA1) (for G(x, ξ) = ϕ(x, |ξ|)) was to assume that Compared to new (VA1), the right-hand side has ϕ(y, t) instead of ϕ(y, t) + 1 but on the other hand the inequality is not assumed for ϕ(y, t) ∈ [0, ω(r)). However, we can show that these are again equivalent if ϕ ∈ Φ w (Ω) is continuous in t. A similar argument applies to the other conditions as well.
Let us show that the old version of (VA1) implies (VA1) with the same ω. Indeed, we need only show that the old condition implies (VA1) when ϕ(y, t) < ω(r) since this is trivial when ϕ(y, t) ∈ [ω(r), |B r | −1 ]. Fix B r and let y ∈ B r . Since ϕ is increasing and continuous, we can find s > t with ϕ(y, s) = ω(r). Then by the old condition, Hence we obtain (VA1) as follows: |ϕ(x, t) − ϕ(y, t)| ϕ(x, s) + ϕ(y, s) (L + 2)ω(r) (L + 2)ω(r)(ϕ(y, t) + 1). 8 For the opposite implication, if ϕ(y, t) ∈ [ω(r) It follows that (VA1) with ω implies the old version of (VA1) with ω 1 2 in place of ω. In summary, under minimal and natural assumptions on G, our conditions are equivalent to previous versions. In particular, from [22,Chapter 7] and [28, Section 8], we know that (A1) is equivalent to logarithmic Hölder continuity in the variable exponent case ϕ(x, t) = t p(x) while (VA1) and (wVA1) hold with ω if and only if p ∈ C 0,ω . For the double phase case ϕ(x, t) = t p + a(x)t q , (VA1) is more or less equivalent to q p < 1 + α n while (wVA1) and (A1) require only q p 1 + α n , where a ∈ C 0,α (Ω). Remark 2.6. Baroni, Colombo and Mingione [2,9] initiated the study of double phase with additional information on the minimizer. Specifically, they considered bounded or Hölder continuous minimizers and showed that the assumption q p 1 + α n can be relaxed in these cases. Ok [37] considered analogous results with Lebesgue integrability information on the minimizer. In [5,24], the generalized Orlicz case with additional information was considered, but only for lower regularity, i.e. Harnack's inequality and Hölder continuity for some γ > 0. It is possible, but non-trivial, to develop the ideas from this paper to prove maximal regularity in the generalized Orlicz case with additional information; we will return to this issue in a follow-up paper [29].

Construction of growth function
In this section, we construct an auxiliary function ϕ ∈ Φ c (Ω) that measures the growth of A : Ω × R n → R n or F : Ω × R n → [0, ∞).
Remark 3.4. In the above definition, we may assume without loss of generality that ϕ(x, ·) ∈ and (Inc) p 1 −1 and (Dec) q 1 −1 with the same p 1 and q 1 .
Proposition 3.5. Every A : Ω × R n → R n satisfying Assumption 1.2 has a growth function ϕ ∈ Φ c (Ω) with constants 1 < p 1 q 1 and 0 < ν Λ depending only on the parameters p, q and L in Assumption 1.2. Specifically, p 1 = p and q 1 q.
From now on, if A : Ω × R n → R n satisfies Assumption 1.2, we take its growth function as the one constructed in the above proposition, hence the constants p 1 , q 1 , ν, Λ in Definition 3.1 depend on the constants p, q, L in Assumption 1.2.
Remark 3.6. Inequality (3.3) with ϕ ′ satisfying (Inc) p−1 and (Dec) q 1 −1 implies the following strict monotonicity condition: Furthermore, lettingξ → 0 in the preceding inequality, we also have the coercivity condition: From now on, we denote and write ϕ ≈ A (−1) as an abbreviation of (3.8). We next consider the continuity hypotheses (wVA1). This assumption is not invariant under equivalence, but we have the following implications for the growth function. The second part of the proposition is a stronger version of (A1), where we notice that the range of ϕ − Br (t) is changed to [ω(r), |B r | −1 ], which is needed later on. Here p ′ denotes the Hölder conjugate exponent of p, i.e. p ′ = p p−1 . Proposition 3.10. Let A : Ω × R n → R n satisfy Assumption 1.2 and ϕ be its growth function. If A (−1) satisfies (wVA1), then for each ε ∈ (0, 1] with ω = ω ε and r ∈ (0, 1], Therefore, by (wVA1) with K = 2c 1 we have that for any x, y ∈ B r and, using A (−1) ≈ ϕ, . Since the implicit constant in the preceding inequality is independent of ε, we obtain the inequality in (2) for ξ with ϕ − Br (|ξ|) ∈ [ω(r), |B r | −1 ). Moreover, the continuity of ϕ in the t variable implies the same inequality for ξ with ϕ − Br (|ξ|) = |B r | −1 . This completes the proof of claim (2).
Let F : Ω × R n → [0, ∞) satisfy Assumption 1.6 and ϕ ∈ Φ c (Ω) be its growth function. By (3.8) In the same way as Propositions 3.9 and 3.10, we can also prove the following.
Proposition 3.14. Let F : Ω×R n → R satisfy Assumption 1.6 and ϕ be its growth function. Then F satisfies (A1) if and only if ϕ satisfies (A1).

Auxiliary regularity results
In this section we collect results on two types of approximating problems, namely nonautonomous problems with Uhlenbeck structure and autonomous problems without Uhlenbeck structure.
With theseĀ andφ, we present regularity results of weak solutions to the following autonomous problem The following results are variations of known regularity results for equations with Orlicz growth, and the proofs are similar to previous ones. In particular, in the Uhlenbeck casē A(ξ) =φ ′ (|ξ|) ξ ξ, we considered them in [28]. Therefore, we outline the proofs and point out differences compared with the references. The first result is local C 1,α -regularity. Lemma 4.4. LetĀ : R n → R n satisfy Assumption 1.2 with A(x, ξ) ≡Ā(ξ) and constants L 1 and 1 < p q, andφ ∈ Φ c be its growth function. Ifū ∈ W 1,φ (B r ) is a weak solution to (divĀ), then Dū ∈ C 0,ᾱ loc (B r , R n ) for someᾱ ∈ (0, 1) with the following estimates: for any B ρ ⊂ B r and any τ ∈ (0, 1), Hereᾱ ∈ (0, 1) and c > 0 depend only on the constants n, p, q and L.
Outline of proof. The proof is almost the same as for the Uhlenbeck case, that is [28, Lemma 4.12] with a proof in [28, Appendix A]. We also refer to [31] for the caseφ(t) = t p . We only need to slightly modify the beginning and approximation parts to the general casē A(ξ).
Note that, by (4.5), b ij (ξ) satisfies Therefore, by following [28, Appendix A], in particular from the paragraph containing (A.7), with the above setting, we can obtain the desired regularity estimates for u ε , where relevant constants are independent of ε. Then using (4.6) and the uniform monotonicity (3.7) with A =Ā ε and ϕ =φ ε , we see thatū ε −→ū in W 1,φ (B r ) as ε → 0. Note that the standard Minty-Browder technique for a monotonicity operator, that is, an operator A which satisfies inequality (3.7) with right hand side 0, implies only weak convergence, but due to the uniform monotonicity we have the strong convergence. In particular, Dū ε (x) converges, up to a subsequence, to Dū(x) almost everywhere in B r .
The next result is a Calderón-Zygmund type estimate in the generalized Orlicz space for non-zero boundary data. Since θ is superlinear, this lemma allows us to transfer regularity from u toū. Lemma 4.7. LetĀ : R n → R n satisfy Assumption 1.2 with A(x, ξ) ≡Ā(ξ) and constants L 1 and 1 < p q, andφ ∈ Φ c be its growth function. Suppose θ ∈ Φ w (B r ) satisfies (A0), (aInc) p θ and (aDec) q θ with constants L θ 1 and 1 < p θ q θ and (A1) with constantL K > 0, and u ∈ W 1,φ (B r ) satisfies´B r θ(x,φ(|Du|)) dx κ for some κ > 0. Ifū ∈ u + W 1,φ 0 (B r ) is a weak solution to (divĀ), then and Br θ(x,φ(|Dū|)) dx c κ Other differences are minor modifications. For a weak solutionv ∈ W 1,φ (B + ρ ) to the above equation, one can prove that for some c > 0 depending only on n, p, q 1 , ν and Λ. This corresponds to [28, (B.13)] in the Uhlenbeck case, which was obtained from the interior counterpart by applying the reflection argument with the odd extension, see the last paragraph in [28]. However, in the general case the reflection argument does not work, and we use a so-called barrier argument. For the proof of the above Lipschitz estimate we refer to [11,Theorem 4.1], see also [10,Theorem 2.2]. Note that Cho [11] proved a Calderón-Zymund type estimate with θ(x, t) ≡ θ(t) for nonhomogeneous equations with the zero boundary condition.
Note that the inequality assumed in the above proposition will be linked to the one in Proposition 3.10(2) and 3.14(2). In the Uhlenbeck case F (x, ξ) = ϕ(x, |ξ|) a suitably mollified version ofφ can be used as the approximation of ϕ [28]. However, as far as we can tell, a similar approach to (5.1) does not work in the general case, so we introduce smooth transitions around t 1 and t 2 with transition functions η i . We deal with A and F in separate subsections.

Approximation for weak solutions.
Let A : Ω × R n → R n satisfy Assumption 1.2, and ϕ ∈ Φ c (Ω) be its growth function. Recall that ϕ ′ satisfies (A0), (Inc) p−1 and (Dec) q 1 −1 . In this subsection, we construct an autonomous nonlinearityĀ : R n → R n with growth functionφ given by (5.1), such thatĀ andφ are comparable with A and ϕ, respectively, in a suitable sense.
Fix any small ball B r = B r (x 0 ) ⋐ Ω satisfying |B r | 1. For t 1 ∈ (0, 1 2 ] and t 2 2 that will be chosen later in the next section, a 1 and a 2 from the definition ofφ, and constants q 1 , ν, Λ given in Proposition 3.5 we define , whereΛ := 2 q 1 −p+3 Λ min{p−1,1} , and η i ∈ C ∞ ([0, ∞)), i = 1, 2, 3, satisfy Figure 1 to assist in following the proof. Proof. Let us start by calculating the derivative of η i (|ξ|)|ξ| p−2 ξ with i = 1, 3: Here ⊗ denotes the tensor product (a i ) i ⊗(b i ) j = (a i b j ) i,j and I n is the n-dimensional identity matrix. We consider the first condition (3.2) of being a growth function withĀ andφ in place of A and ϕ. When |ξ| < 2t 1 , we use |η ′ Similarly, we conclude that |Ā 3 (ξ)| + |ξ||D ξĀ3 (ξ)| For the ellipticity condition (3.3), we consider four cases: 0 < |ξ| t 1 , t 1 < |ξ| 2t 1 , t 2 /2 < |ξ| t 2 and t 2 < |ξ| 2t 2 . The strategy is to get the ellipticity condition by the same condition ofĀ 1 andĀ 2 for first and third cases, respectively. In the second and fourth cases one term of D ξĀ is non-negative and the other is non-positive (since η ′ 1 or η ′ 2 is non-positive), so we have to show that the non-positive term can be absorbed in the non-negative one based on more precise estimates.

5.2.
Approximation for minimizers. Let F : Ω × R n → [0, ∞) satisfy Assumption 1.6 with constants L 1 and 1 < p < q, and ϕ ∈ Φ c (Ω) be its growth function. We construct an autonomous functionF : R n → [0, ∞) with growth functionφ given in (5.1). The construction is similar to, yet more delicate than, that of the previous subsection. The added difficulty comes from the fact that we need to differentiate F twice, which makes controlling the approximation more challenging. Fix any small ball B r = B r (x 0 ) ⋐ Ω with r ∈ (0, 1) satisfying |B r | 1. For t 1 ∈ (0, 1 2 ], t 2 2,ν ≪ 1 andΛ ≫ 1 that will be chosen later, we define 10,

10.
Observe that η 1 and η 2 are analogous to their namesakes in the previous subsection, but η 3 behaves somewhat differently. The reason is that if η 3 is forced to be constant from 2t 2 19 onward as in the previous subsection, then it is not possible to control derivatives up to order 2 in an appropriate manner. We again note that (5.7) η i (t) + t|η ′ i (t)| + t 2 |η ′′ i (t)| C for i = 1, 2, 3 and all t 0, and that F (ξ) = F (x 0 , ξ) whenever 2t 1 |ξ| t 2 2 . Lemma 5.8. If ϕ is a growth function of F , andφ andF are their autonomous approximations as in (5.1) and (5.6), thenφ is a growth function ofF .

Regularity of weak solutions and minimizers
In this section we prove the main theorems stated in the introduction. We consider either A : Ω × R n → R n satisfying Assumption 1.2 with A (−1) satisfying (wVA1) or F : Ω × R n → R satisfying Assumption 1.6 and (wVA1). Note that the parameters p, q and L are from Assumption 1.2 or 1.6, and the parameterL is given byL K in (wVA1) with where c 1 and c 2 are from Propositions 3.10 and 3.14. Let ϕ be a growth function of A or F . Note that by Proposition 3.10 or 3.14, A (−1) or F satisfies (A1) when K = 1 withL 1 depending only on n, p, q, L andL, hence so does ϕ. Let u ∈ W 1,ϕ loc (Ω) be a local weak solution to (divA) or a local minimizer of (min F ). Then, by Theorem 4.1 along with Remark 4.2, we have ϕ(·, |Du|) ∈ L 1+σ loc (Ω) for some σ = σ(n, p, q, L,L) ∈ (0, 1). With this σ, we fix the parameters ε and ω = ω ε from (wVA1) by We also fix Ω ′ ⋐ Ω, and consider any B 2r ⊂ Ω ′ with r ∈ (0, 1 2 ) satisfying Note that using Hölder's inequality we have so that Note that this allows us to take advantage of the higher integrability estimate in Theorem 4.1.

which implies (3) sinceφ is strictly increasing.
We next estimate the difference of the gradient between u andū in the L 1 -sense. This generalizes Lemma 6.2 and Corollary 6.3 of [28] to the non-Uhlenbeck case. for some c 1 and γ ∈ (0, 1) depending only on n, p, q, L andL.
Proof of Theorem 1.3. We can prove the theorem using Lemmas 4.4, 6.6 and 6.7 using a standard iteration argument. The proof is exactly the same as [28,Theorem 7.2], see also [1,2], so we omit it.
6.2. Regularity for minimizers. In this subsection we prove Theorem 1.7. The method is almost the same as for Theorem 1.3 except for the comparison step. Hence we will take advantage of many parts from the previous subsection. We assume F : Ω × R n → R satisfies Assumption 1.6 and (wVA1), and consider a local minimizer u ∈ W 1,ϕ loc (Ω) of (min F ). Recall that the minimizer u is also a weak solution to (divA) with A = D ξ F , but (wVA1) of F does not imply (wVA1) of A := D ξ F . On the other hand, by Proposition 3.14(1) we have . Moreover, by the Proposition 3.14(2), the condition of Proposition 5.2 holds withL =L(n, p, q, L,L).
Proof of Theorem 1.7. The proof is exactly same as that of Theorem 1.3, with Lemma 6.7 replaced by Lemma 6.9.

Examples
We present two known, important nonstandard growth problems, and show that they are special cases of Theorem 1.3 or 1.7. Various other examples of growth functions together with references about regularity results for related equations and minimizing problems can be found in [28].