Regularity theory for non-autonomous problems with a priori assumptions

We study weak solutions and minimizers u of the non-autonomous problems divA(x,Du)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {div}} A(x, Du)=0$$\end{document} and minv∫ΩF(x,Dv)dx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\min _v \int _\Omega F(x,Dv)\,dx$$\end{document} 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,ξ)=|ξ|p+a(x)|ξ|q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(x, \xi )=|\xi |^p + a(x)|\xi |^q$$\end{document}. 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.


Introduction
We consider the divergence form, quasilinear elliptic equation 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 (ξ) := |ξ| p , p ∈ (1, ∞) and the p-Laplace equation with A(ξ) := |ξ| p−2 ξ.The maximal regularity of weak solutions already to the p-Laplace equation when p = 2 is C 1,α for some α ∈ (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, |ξ| p F (ξ) |ξ| q + 1 and 1 < p q, provided that 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, ξ) := |ξ| p + a(x)|ξ| q with 1 < p q and a : Ω → [0, L ω ], 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 α of a and established maximal regularity of the minimizer u in the following three cases of a priori information: (ap1) u ∈ W 1,p (Ω) and q − p p n α (ap2) u ∈ L ∞ (Ω) and q − p α (ap3) u ∈ C 0,γ (Ω) and q − p < 1  1−γ α 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 ∈ L s * (Ω) and q − p s n α In view of the Sobolev embedding when s < n, s = n and s > n, this suggests the unifying, albeit slightly stronger, assumption u ∈ W 1,s (Ω) and q − p s n α.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, ξ) = |ξ| 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, ξ) = F (x, |ξ|) 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 ∈ W 1,ϕ (Ω) 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: 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 ∞ .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) q(x) be Hölder continuous and bounded away from 1 and ∞, and a ∈ C 0,αa and b ∈ C 0,α b be non-negative and bounded.Assume that F (x, ξ) = f (x, |ξ|) equals one of following functions with corresponding additional conditions hold: Perturbed double phase t p + a(x)t q log(e + t) α a q − p Triple phase Variable exponent double phase t p(x) + a(x)t q(x) α a (x) q(x) − p(x) Remark 1.2.The previous corollary holds also with the weaker, but more difficult to check, assumption u ∈ W 1,1 loc (Ω) ∩ BMO(Ω).Without the additional a priori information u ∈ BMO(Ω), the additional conditions are These are stronger assumptions when p < n as expected, since if p n, then W 1,p loc (Ω) ⊂ BMO loc (Ω), so the a priori assumption u ∈ BMO(Ω) 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 ϕ(t) p(x) or ϕ(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 (A1-n) holds if and only if the (A1) condition holds, see Section 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 ϕ(t) + a(x)ψ(t), where a ∈ C 0,λ and ψ/ϕ is almost increasing, but the conditions get a bit messy.The main condition is that for each ε > 0 there exists β > 0 such that 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 ∞ or C 0,γ , 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 : Ω×R n → R n from (div A) and F : Ω × R n → 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 Section 3, we consider lower order regularity in cases of generalized Orlicz growth and a priori information.We prove C 0,α -regularity for some α ∈ (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 Section 5 we prove the main results, maximal regularity of weak solutions and minimizers for cases (ap2)-(ap4).We prove C 0,α -regularity for every α ∈ (0, 1) and C 1,α -regularity for some α ∈ (0, 1) assuming a priori C 0,γ -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 (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 Section 2 by recalling notation, definitions and basic results on generalized Orlicz spaces.

Preliminaries and notation
Throughout the paper we always assume that Ω is a bounded domain in R n with n 2. For x 0 ∈ 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 Let f, g : E → R be measurable in E ⊂ R n .We denote the integral average of f over Similarly, we define an almost decreasing or decreasing function.We write f g, f ≈ g and f ≃ g if there exists C 1 such that f (y) Cg(y), C −1 f (y) g(y) Cf (y) and f (C −1 y) g(y) f (Cy) for all y ∈ E, respectively.We use c as a generic constant whose value may change between appearances.
We next introduce classes of Φ-functions and generalized Orlicz spaces following [33].We are mainly interested in convex functions for minimization problems and related PDEs, but the class Φ w (Ω) is very useful for approximating functionals.
the relation ≃ is weaker than ≈, but they are equivalent if ϕ and ψ satisfy (aDec).We write The (left-continuous) inverse function with respect to t is defined by If ϕ is strictly increasing and continuous in t, then this is just the normal inverse.We define the conjugate function of ϕ ∈ Φ w (Ω) by The definition directly implies Young's inequality ts ϕ(x, t) + ϕ * (x, s) for all s, t 0.
with constant depending on L and γ.

Lower regularity with a priori assumptions
Continuity assumptions.The condition (A1), 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 (A1)" condition, denoted (VA1), and a weak vanishing version, (wVA1) 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 ∈ W 1,ϕ (Ω), called Case (ap1) in the introduction.The (A1-n) and (A1-ψ) conditions for a priori energy assumptions were developed in [36] and [11] for functions in L ∞ and W 1,ψ , respectively.Here we generalize and unify all the conditions for a priori information; the most important one for this article is (VA1-s).
Table 1.Examples of sufficient conditions in special cases.
The next example shows the relevance of the parameter s in (A1-s) in the double phase case.Similar relations for other cases are summarized in Table 1.
If q−p s n α, then ϕ 2 satisfies (A1-s) and (wVA1-s) with ω ε (r) = cr {α− n(q−p) α } .Note that these conditions can hold for 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 α < 1.We will use the elementary inequality which holds for any a, b 0 and δ ∈ (0, 1]; here c α > 0 is a constant depending on α.The second inequality (α > 1) follows from Young's inequality applied to the right-hand side of b α − a α αb α−1 (b − a) when b a 0. Suppose that q − p s n α and let x, y ∈ B r with r ∈ (0, 1] and t ∈ (0, |B r | − 1 s(1+ε) ] with ε 0. Applying the preceding inequality with a = a(x) and b = a(y), we obtain that |ϕ 2 (x, t) − ϕ 2 (y, t)| r α t q = r α t q−p t p r α− n(q−p) (1+ε)s ϕ 2 (y, t) when 0 < α 1; in the case α > 1, we choose δ := r {α− n(q−p) α and find that These inequalities imply the desired (A1-s), (wVA1-s) and (VA1-s)-conditions.
Let us show how to use the smallness of ω to obtain the inequality from (VA1-s) 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 ξ in the condition is independent of x, so the result does not generalize to (VA1-ψ) easily, unless ψ(x, t) = ψ(t).
Proof.Note by the concavity of log that t log 2 log(1 + t) t when t ∈ [0, 1], and set where ⌊τ ⌋ is the largest integer less than or equal to τ ∈ R. Suppose x, y ∈ B r ∩ Ω and . 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 We use this estimate with the triangle inequality and (VA1-s): This gives the desired estimate, since by the definition of k, 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 n.This approach is inspired by [11].
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 (A1-s) is replaced by (A1-s ′ ) 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 ⊂ R 2 , k ∈ {1, 2, 3, 4}.Let η : [0, 4] → [0, 1] be the piecewise linear, 3-Lipschitz function with η [ 1 Thus a equals 0 in Q 2 and Q 4 and a(x) = |x| α in the sectors with π 6 < θ < π 3 in Q 1 and with 7π 6 < θ < 4π 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 : R 2 → R which equals 1 in Q 1 , −1 in Q 3 and is linear in the polar coordinate θ in Q 2 and Q 4 .By symmetry, u Br = 0 for every ball B r centered at the origin and v := 1 r |u − (u) Br | = 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 4  πr .For a constant k > 0 we estimate, based on the sectors in Q 1 and Q 3 , Br H(x, kv) dx 1 3r 2 ˆr 0 s 1+α ( k r ) q ds = 1 3(2+α) r α−q k q and, since the support of the derivative is If the modular Poincaré inequality from Theorem 3.4 holds with θ 0 = 1 (the weakest relevant case), then r α−q k q cr −p k p + c so that r α−(q−p) k q−p c + cr p k −p .
Suppose that we want the constant in the inequality to depend on the L s * -norm of ku.
We calculate ku L s * (Br) = ckr 2/s * .Thus k = cr −2/s * and so r α−(q−p) k q−p ≈ r α−( 2 s * +1)(q−p) = r α− 2 s (q−p) .This remains bounded as r → 0 when α 2 s (q − p) which is exactly the (A1-s) 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 ∞ -norm.Similarly, we see that if the constant is allowed to depend on ρ H (|∇u|), then α 2 p (q − p) which is (A1).Unfortunately, Case (1) is not covered, since the counter-example is discontinuous.
The constants σ and c depend only on n, p, q, L, L ω , Q and for every ball B 2ρ ⊂ B 2r , where θ 1 and c 1 depend on the parameters listed in the statement.By Gehring's Lemma (e.g.[31,Theorem 6.6]), there exists σ > 0 depending on θ 1 condition which is a variant of (p, q)-growth of F .The latter is equivalent to the existence of L 1 such that L for all x ∈ Ω and t > 0.
Note that all examples in Table 1 satisfy this condition, but not L for all t > 0, which holds for F (x, ξ) ≈ a(x)ψ(|ξ|) with 0 < ν a L. We remark (4.2) and (4.3) are called the pointwise and global uniform ellipticity condition.Here, "uniform" is concerned with the variable |ξ|.For more discussion about this uniform ellipticity condition, we refer to [22].
for every x ∈ Ω and ϕ ′ satisfies (A0), (Inc) p 1 −1 and (Dec) q 1 −1 as well as for all x ∈ Ω and ξ, ξ ∈ R n \ {0}; in the case of F we assume that the the inequalities hold for A := D ξ F .
In this paper we always use the growth function from the following proposition.Thus the additional parameters p 1 , q 1 , ν and Λ only depend on the original parameters p, q and L. Proposition 4.5 (Proposition 3.3, [42]).Every A : Ω×R n → R n and F : Ω×R → [0, ∞) with quasi-isotropic (p, q)-growth has a growth function ϕ ∈ Φ c (Ω) with p 1 = p, q 1 q, ν and Λ depending only on p, q and L. By Proposition 2.3, the growth function ϕ satisfies (A0), (Inc) p , (Dec) q 1 as well as ϕ * (x, ϕ ′ (x, t)) ϕ ′ (x, t)t ≈ ϕ(x, t).Furthemore, by [42,Remark 3.4] we have the strict monotonicity condition as well as the equivalences where the implicit constants depend on only p, q and L.
Let A : Ω × R n → R n have quasi-isotropic (p, q)-growth and growth function ϕ and let u ∈ W 1,1 loc (Ω) be a weak solution to (div A).We showed in [42, Section 4.1] that u is a quasiminimizer of the ϕ-energy for some Q = Q(p, q, L) 1; in the reference we assumed that smooth functions are dense in the Sobolev space W 1,ϕ , which is reasonable if ϕ satisfies (A1).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 Section 3 to weak solutions.
We next show how the (A1)-type condition of A or F transfers to the growth function ϕ.Based on Part (2), we can say that (VA1-ψ) and (wVA1-ψ) are weaker in the minimization case than in the PDE case.This justifies studying minimizers separately.
(2) If A := D ξ F satisfies (VA1-ψ), then so does F , with the same ω up to a constant depending on p, q and L. (3) If A or F satisfies (VA1-s), s > 0, and θ ∈ (0, 1], then The constant c > 0 depends only on p, q and L.
We omit the proof of (2) which is essentially the same as [42,Proposition 3.8].
When considering equations with nonlinearity A it is natural to assume (VA1-ψ) for the function A (−1) (x, ξ) := |ξ|A(x, ξ).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 : Ω × R n → R n have quasi-isotropic (p, q)-growth.Then A satisfies (VA1-ψ) if and only if A (−1) does.If the original modulus continuity is ω, then the new modulus continuity can be taken as c ω 1 p ′ for some c = c(p, q, L, L ω ) > 0. Dividing both sides by |ξ| gives the desired estimate.
Remark 4.10.In [42,Proposition 3.8] we assumed that A (−1) satisfies (wVA1) 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 Ā : R n → R n via its trivial extension Ā(x, ξ) := Ā(ξ).It has a growth function φ ∈ Φ c ∩ C 1 ([0, ∞)), cf.[42].For such Ā and φ, we present regularity results of weak solutions to Note that we use the bar-symbol to indicate the autonomous versions of A, F , ϕ and corresponding solutions or minimizers u.In the Uhlenbeck case Ā(ξ) = φ′ (|ξ|) ξ ξ, we proved the next result in [41] and in [42] we sketched how to extend the proof to the quasiisotropic case.Since the result is independent of the (A1)-type assumptions, it applies directly also to this paper.Lemma 4.11 (C 1,α -regularity, Lemma 4.4, [42]).Let Ā : R n → R n have quasi-isotropic (p, q)-growth and φ ∈ Φ c be its growth function.If ū ∈ W 1, φ(B r ) is a weak solution to (div Ā), then Dū ∈ C 0, ᾱ loc (B r , R n ) for some ᾱ ∈ (0, 1) with the following estimates: for every B ρ ⊂ B r and τ ∈ (0, 1).Here ᾱ and c > 0 depend only on n, p, q and L.
Proposition 5.2.Let t K , ϕ and φ be as above.Suppose that ϕ + Br (t) L K ϕ − Br (t) for some L K 0 and all t ∈ [1, t K ].
(3) φ(t) q 1 p L K ϕ(x, t) + L for all (x, t) ∈ B r × [0, ∞) and W 1,ϕ (B r ) ⊂ W 1, φ(B r ).Proof.Parts (1) and ( 2) follow directly from the definition of φ and the inclusion in (3) follows from the inequality.If t 1, then the inequality in (3) follows from (A0) and when t ∈ [1, t K ], it follows from ϕ + Br (t) L K ϕ − Br (t) and ( 2).If t > t K , we calculate φ(t) = ϕ(x 0 , t K ) + ˆt t K ϕ ′ (x 0 , t K ) We prove our main theorem on Hölder continuous weak solutions to (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 = α 1−γ with a ∈ V C 0,α .
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 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,
v)) Br and by the modular Poincaré inequality in the Orlicz space L ψ [33, Corollary 7.4.1],(ψ(v)) Br can be estimated by the first term on the right-hand side.Combined with the inequality from the previous paragraph, this gives the claim.