Higher integrability for variational integrals with non-standard growth

We consider autonomous integral functionals of the form F[u]:=∫Ωf(Du)dxwhereu:Ω→RN,N≥1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathcal {F}}[u]:=\int _\varOmega f(D u)\,dx \quad \text{ where } u:\varOmega \rightarrow {\mathbb {R}}^N, N\ge 1, \end{aligned}$$\end{document}where the convex integrand f satisfies controlled (p, q)-growth conditions. We establish higher gradient integrability and partial regularity for minimizers of F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {F}}$$\end{document} assuming qp<1+2n-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{q}{p}<1+\frac{2}{n-1}$$\end{document}, n≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 3$$\end{document}. This improves earlier results valid under the more restrictive assumption qp<1+2n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{q}{p}<1+\frac{2}{n}$$\end{document}.


Introduction
In this note, we study regularity properties of local minimizers of integral functionals where Ω ⊂ R n , n ≥ 3, is a bounded domain, u : Ω → R N , N ≥ 1 and f : R N ×n → R is a sufficiently smooth integrand satisfying ( p, q)-growth of the form Assumption 1 There exist 0 < ν ≤ L < ∞ such that f ∈ C 2 (R N ×n ) satisfies for all z, ξ ∈ R N ×n Regularity properties of local minimizers of (1) in the case p = q are classical, see, e.g., [24]. A systematic regularity theory in the case p < q was initiated by Marcellini in [27,28], see [31] for an overview (for a more up-to-date overview see the introduction in [30]). In particular, Marcellini [29] proves (among other things): (A) If N = 1, 2 ≤ p < q and q p < 1 + 2 n , then every local minimizer u ∈ W 1, p loc (Ω) of (1) satisfies u ∈ W 1,∞ loc (Ω). Local boundedness of the gradient implies that the non-standard growth of f and ∂ 2 f in (1) becomes irrelevant and higher regularity (depending on the smoothness of f ) follows by standard arguments, see e.g. [27,Chapter 7].
Only very recently, Bella and the author improved in [6] the result (A) in the sense that 'n' in the assumption on the ratio q p can be replaced by 'n −1' for n ≥ 3 (to be precise, [6] considers the non-degenerate version (4) of (2)). The argument in [6] relies on scalar techniques, e.g., Moser-iteration type arguments, and thus cannot be extended to the vectorial case N > 1.
To the best of our knowledge, there is no improvement of (B) with respect to the relation between the exponents p, q and the dimension n available in the literature. Here we provide such an improvement for n ≥ 3.
Before we state the results, we recall a standard notion of local minimizer in the context of functionals with ( p, q)-growth Definition 1 We call u ∈ W 1,1 loc (Ω) a local minimizer of F given in (1) iff The main result of the present paper is Let u ∈ W 1,1 loc (Ω, R N ) be a local minimizer of the functional F given in (1).
Higher gradient integrability is a first step in the regularity theory for integral functionals with ( p, q)-growth, see [7,11,19,20] for further higher integrability results under ( p, q)conditions. Clearly, we cannot expect to improve from W 1,q loc to W 1,∞ loc for N > 1, since this even fails in the classic setting p = q, see [34]. Direct consequences of Theorem 2 are higher differentiability and a further improvement in gradient integrability in the form: (i) (Higher differentiability). In the situation of Theorem 2 it holds |∇u| p−2 2 ∇u ∈ W 1,2 loc (Ω), see Theorem 5. (ii) (Higher integrability). Sobolev inequality and (i) imply ∇u ∈ L κ p loc (Ω, R N ×n ) with κ = n n−2 . Note that κ p > q provided q p < 1 + 2 n−2 . A further, on first glance less direct, consequence of Theorem 2 is partial regularity of minimizers of (1), see, e.g., [1,7,10,32], for partial regularity results under ( p, q)-conditons. For this, we slightly strengthen the assumptions on the integrand and suppose In [7], Bildhauer and Fuchs prove partial regularity under Assumption 3 with q p < 1 + 2 n ( [7] contains also more general conditions including, e.g., the subquadratic case). Here we show

Then, there exists an open set
We do not know if (3) in Theorems 2 and 4 is optimal. Classic counterexamples in the scalar case N = 1, see, e.g., [23,28], show that local boundedness of minimizers can fail if q p is to large depending on the dimension n. In fact, [28, Theorem 6.1] and the recent boundedness result [26] show that 1 p − 1 q ≤ 1 n−1 is the sharp condition ensuring local boundedness in the scalar case N = 1 (for sharp results under additional structure assumptions, see, e.g., [14,22]).
For non-autonomous functionals, i.e., Ω f (x, Du) dx, rather precise sufficiently & necessary conditions are established in [20], where the conditions on p, q and n has to be balanced with the (Hölder)-regularity in space of the integrand. However, if the integrand is sufficiently smooth in space, the regularity theory in the non-autonomous case essentially coincides with the autonomous case, see [10]. Currently, regularity theory for non-autonomous integrands with non-standard growth, e.g. p(x)-Laplacian or double phase functionals are a very active field of research, see, e.g., [2,12,13,[15][16][17]25,33].
Coming back to autonomous integral functionals: In [11] higher gradient integrability is proven assuming so-called 'natural' growth conditions, i.e., no upper bound assumption on ∂ 2 f , under the relation q p < 1 + 1 n−1 . Moreover, in two dimensions we cannot improve the previous results on higher differentiability and partial regularity of, e.g., [7,18], see [8] for a full regularity result under Assumption 3 with n = 2 and q p < 2. Finally, we mention the recent paper [3] where optimal Lipschitz-estimates with respect to a right-hand side are proven for functionals with ( p, q)-growth.
Let us briefly describe the main idea in the proof of Theorem 2 and from where our improvement compared to earlier results comes from. The main point is to obtain suitable a priori estimates for minimizers that may already be in W 1,q loc (Ω, R N ). The claim then follows by a known regularization and approximation procedure, see, e.g., [18]. For minimizers v ∈ W 1,q loc (Ω, R N ) a Caccioppoli-type inequality is valid for all sufficiently smooth cut-off functions η, see Lemma 1. Very formally, the Caccioppoli inequality (5) can be combined with Sobolev inequality and a simple interpolation inequality to obtain The Dv L κ p -factor on the right-hand side can be absorbed provided we have qθ p < 1, but this is precisely the 'old' ( p, q)-condition q p < 1 + 2 n , this type of argument was previously rigorously implemented in, e.g., [7,19]. Our improvement comes from choosing a cut-of function η in (5) that is optimized with respect to v, which enables us to use Sobolev inequality on n − 1-dimensional spheres wich gives the desired improvement, see Sect. 3. This idea has its origin in joint works with Bella [4,5] on linear non-uniformly elliptic equations.
With Theorem 2 at hand, we can follows the arguments of [7] almost verbatim to prove Theorem 4. In Sect. 4, we sketch (following [7]) a corresponding ε-regularity result from which Theorem 4 follows by standard methods.

Preliminary results
In this section, we gather some known facts. We begin with a well-known higher differentiability result for minimizers of (1) under the assumption that u ∈ W 1,q loc (Ω, R N ): The Lemma 1 is known, see e.g. [7,18,28]. Since we did not find a precise reference for estimate (6), we included a prove here following essentially the argument of [18].

Proof of Lemma 1
Without loss of generality, we suppose ν = 1 the general case ν > 0 follows by replacing f with f /ν (and thus L with L/ν). Throughout the proof, we write if ≤ holds up to a multiplicative constant depending only on n, N , p and q.
Thanks to the assumption v ∈ W (for this we use that the convexity and growth conditions of f imply |∂ f (z)| ≤ c(1 + |z| q−1 ) for some c = c(L, n, N , q, ) < ∞). Next, we use the difference quotient method, to differentiate the above equation: For s ∈ {1, . . . , n}, we consider the difference quotient operator t=0 , the fundamental theorem of calculus yields where we use τ h,s Q = Qe s . Youngs inequality yields where Combining (8), (9) with the assumptions on ∂ 2 f , see (2) Estimate (10), the fact v ∈ W 1,q loc (Ω) and the arbitrariness of η ∈ C 1 c (Ω) and s ∈ {1, . . . , n} yield |Dv| the desired estimate (6) follows by summing over s.
Next, we state a higher differentiability result under the more restrictive Assumption 3 which will be used in the proof of Theorem 4.
Let v ∈ W 1,q loc (Ω, R N ) be a local minimizer of the functional F given in (1). Then, h := A variation of Lemma 2 can be found in [7] and we only sketch the proof.

Proof of Lemma 2
With the same argument as in the proof of Lemma 1 but using (4) instead of (2), we obtain v ∈ W 2,2 loc (Ω, R N ) and the Caccioppoli inequality Formally, the chain-rule implies where c = c(n, p) < ∞, and the claimed estimate (11) follows from (12) and (13). In general, we are not allowed to use the chain rule, but the above reasoning can be made rigorous: For h m we are allowed to use the chain-rule and (12) Then for every δ ∈ (0, 1] For convenience of the reader we include a short proof of Lemma 3 (14) follows directly by minimizing among radial symmetric cut-off functions. Indeed, we obviously have for every ε ≥ 0

Higher integrability -Proof of Theorem 2
In this section, we prove the following higher integrability and differentiability result which clearly contains Theorem 2 Theorem 5 Let Ω ⊂ R n , n ≥ 2, and suppose Assumption 1 is satisfied with 2 ≤ p < q < ∞ such that q p < 1 + min{ 2 n−1 , 1}. Let u ∈ W 1,1 loc (Ω, R N ) be a local minimizer of the functional F given in (1). Then, u ∈ W 1,q loc (Ω, R N ) and |Du| where Proof of Theorem 5 Without loss of generality, we suppose ν = 1 the general case ν > 0 follows by replacing f with f /ν. Throughout the proof, we write if ≤ holds up to a multiplicative constant depending only on L, n, N , p and q.
Following, e.g., [7,18,19], we consider the perturbed integral functionals We then derive suitable a priori higher differentiability and integrability estimates for local minimizers of F λ that are independent of λ ∈ (0, 1). The claim then follows with help of a by now standard double approximation procedure in spirit of [18].
Appealing to the bounds (35), (36) and lower semicontinuity, we obtain Inequality (38), strict convexity of f and the fact w ∈ u + W 1, p 0 (B 1 ) imply w = u and thus the claimed estimate (31) is a consequence of (37).

Partial regularity -Proof of Theorem 4
Theorem 4 follows from, the higher integrability statement Theorem 2, the ε-regularity statement of Lemma 4 below and a well-known iteration argument.

Lemma 4
Let Ω ⊂ R n , n ≥ 3, and suppose Assumption 3 is satisfied with 2 ≤ p < q < ∞ such that q p < 1 + 2 n−1 . Fix M > 0. There exists C * = C * (n, N , p, q, L ν , M) ∈ [1, ∞) such that for every τ ∈ (0, 1 4 ) there exists ε = ε(M, τ ) > 0 such that the following is true: Let u ∈ W 1,1 loc (Ω, R N ) be a local minimizer of the functional F given in (1). Suppose for some ball B r (x) Ω where we use the shorthand (w) x,r := − B r (x) w dy, and With the higher integrability of Theorem 5 and the Caccioppoli inequality of Lemma 2 at hand, we can prove Lemma 4 following almost verbatim the proof of the corresponding result [7, Lemma 4.1], which contain the statement of Lemma 4 under the assumption q p < 1 + 2 n (note that in [7] somewhat more general growth conditions including also the case 1 < p < q are considered). Thus, we only sketch the argument.

Proof of Lemma 4 Fix
where C * is chosen below. We consider the sequence of rescaled functions given by The definition of v m yields Assumptions (39) and (40) imply The bound (42) together with (41) imply the existence of v ∈ W 1,2 (B 1 , R N ) such that, up to extracting a further subsequence, see, e.g., [21] or [7,Proposition 4.2]. Standard estimates for linear elliptic systems with constant coefficients imply v ∈ C ∞ loc (B 1 , R N ) and existence of C * * < ∞ depending only on n, N and the ellipticity contrast of ∂ 2 f (A) (and thus on L ν , p, q, and M) such that Choosing C * = 2C * * we obtain a contradiction between (43) and (44)  Hence, using assumption q p < 1 + 2 n−1 (and thus 2q p < 2n n−2 ), we obtain for every ρ ∈ (0, 1) Funding Open Access funding enabled and organized by Projekt DEAL.
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