Higher integrability for variational integrals with non-standard growth

We consider autonomous integral functionals of the form $\mathcal F[u]:=\int_\Omega f(D u)\,dx$ with $u:\Omega\to\mathbb R^N$ $N\geq1$, where the convex integrand $f$ satisfies controlled $(p,q)$-growth conditions. We establish higher gradient integrability and partial regularity for minimizers of $\mathcal F$ assuming $\frac{q}p<1+\frac2{n-1}$, $n\geq3$. This improves earlier results valid under the more restrictive assumption $\frac{q}p<1+\frac2{n}$.

Regularity properties of local minimizers of (1) in the case p = q are classical, see, e.g., [23]. A systematic regularity theory in the case p < q was initiated by Marcellini in [25,26], see [27] for an overview. In particular, Marcellini [26] proves (among other things): (A) If N = 1, 2 ≤ p < q and q p < 1 + 2 n−2 if n ≥ 3, then every local minimizer u ∈ W 1,q 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. [25,Chapter 7]. However, the W 1,q loc (Ω)-assumption on u in (A) is problematic: a priori we can only expect that minimizers of (1) are in the larger space u ∈ W 1,p loc (Ω). Hence, a first important step in the regularity theory for integral functionals with (p, q)-growth is to improve gradient integrability for minimizers of (1). In [17], Esposito, Leonetti and Mingione showed (B) If 2 ≤ p < q and q p < 1 + 2 n , then every local minimizer u ∈ W 1,p loc (Ω, R N ) of (1) satisfies u ∈ W 1,q loc (Ω, R N ). The combination of (A) and (B) yields unconditional Lipschitz-regularity for minimizers of (1) in the scalar case under assumption q p < 1 + 2 n , see [3] for a recent extension which includes in an optimal way a right-hand side. Only very recently, Bella and the author improved in [6] the results (A) and (B) (in the case N = 1) 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, [6,25,26] consider 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. In this paper, we extend the gradient integrability result of [6] to the vectorial case N > 1. 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 The main result of the present paper is Theorem 1. Let Ω ⊂ R n , n ≥ 3, and suppose Assumption 1 is satisfied with 2 ≤ p < q < ∞ such that 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 ). As mentioned above, higher gradient integrability is a first step in the regularity theory for integral functionals with (p, q)-growth, see [11,18,19,7] 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 [30]. Direct consequences of Theorem 1 are higher differentiability and a further improvement in gradient integrability in the form: (i) (Higher differentiability). In the situation of Theorem 1 it holds |∇u| p−2 2 ∇u ∈ W 1,2 loc (Ω), see Theorem 3. (ii) (Higher integrability). Sobolev inequality and (i) imply ∇u ∈ L κp loc (Ω, R N ×n ) with κ = n n−2 . Note that κp > q provided q p < 1 + 2 n−2 . A further, on first glance less direct, consequence of Theorem 1 is partial regularity of minimizers of (1), see, e.g., [1,7,10,28], for partial regularity results under (p, q)-conditons. For this, we slightly strengthen the assumptions on the integrand and suppose Assumption 2. There exist 0 < ν ≤ L < ∞ such that f ∈ C 2 (R N ×n ) satisfies for all z, ξ ∈ R N ×n (4) ν|z| In [7], Bildhauer and Fuchs prove partial regularity under Assumption 2 with q p < 1 + 2 n ( [7] contains also more general conditions including, e.g., the subquadratic case). Here we show Theorem 2. Let Ω ⊂ R n , n ≥ 3, and suppose Assumption 2 is satisfied with 2 ≤ p < q < ∞ such that (3). Let u ∈ W 1,1 loc (Ω, R N ) be a local minimizer of the functional F given in (1). Then, there exists an open set Ω 0 ⊂ Ω with |Ω \ Ω 0 | = 0 such that ∇u ∈ C 0,α (Ω 0 , R N ×n ) for each 0 < α < 1.
We do not know if (3) in Theorem 1 and 2 is optimal. Classic counterexamples in the scalar case N = 1, see, e.g., [22,26], show that local boundedness of minimizers can fail if q p is to large depending on the dimension n. In fact, [26, Theorem 6.1] and the recent boundedness result [24] 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., [14,21]). For non-autonomous functionals, i.e.,´Ω f (x, Du) dx, rather precise sufficiently & necessary conditions are established in [19], 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., [2,12,13,15,16,29].
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., [7,17], see [8] for a full regularity result under Assumption 2 with n = 2 and q p < 2.
Let us briefly describe the main idea in the proof of Theorem 1 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., [17]. 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 κp -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., [7,18]. 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 Section 3. This idea has its origin in joint works with Bella [4,5] on linear non-uniformly elliptic equations. With Theorem 1 at hand, we can follows the arguments of [7] almost verbatim to prove Theorem 2. In Section 4, we sketch (following [7]) a corresponding ε-regularity result from which Theorem 2 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 ): Lemma 1. Let Ω ⊂ R n , n ≥ 2, and suppose Assumption 1 is satisfied with 2 ≤ p < q < ∞. Let v ∈ W 1,q loc (Ω, R N ) be a local minimizer of the functional F given in (1). Then, |Dv| The Lemma 1 is known, see e.g. [7,17,26]. Since we did not find a precise reference for estimate (6), we included a prove here following essentially the argument of [17].
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 1,q loc (Ω, R N ), the minimizer v satisfies the Euler-Largrange equation (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 yieldŝ Ωˆ1 0 where we use τ h,s ℓ Q = Qe s . Youngs inequality yields Combining (8), (9) with the assumptions on ∂ 2 f , see (2), with the elementary estimate 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| p−2 2 Dv ∈ W 1,2 loc (Ω). Sending h to zero in (10), we obtain the desired estimate (6) follows by summing over s.
Next, we state a higher differentiability result under the more restrictive Assumption 2 which will be used in the proof of Theorem 2.
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 where c = c( L ν , n, N, p, q) < ∞. 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)  The following technical lemma is contained in [6] (see also [4, proof of Lemma 2.1, Step 1]) and plays a key role in the proof of Theorem 1 Then for every δ ∈ (0, 1] For convenience of the reader we include a short proof of Lemma 3 Proof of Lemma 3. Estimate (14) follows directly by minimizing among radial symmetric cut-off functions. Indeed, we obviously have for every ε ≥ 0 For ε > 0, the one-dimensional minimization problem J 1d,ε can be solved explicitly and we obtain To see (15), we observe that using the assumption v ∈ L 1 (B σ ) and a simple approximation argument we can replace η ∈ C 1 (ρ, σ) with η ∈ W 1,∞ (ρ, σ) in the definition of J 1d,ε . Let η : [ρ, σ] → [0, ∞) be given by Clearly, η ∈ W 1,∞ (ρ, σ) (since b ≥ ε > 0), η(ρ) = 1, η(σ) = 0, and thus The reverse inequality follows by Hölder's inequality. Next, we deduce (14) from (15) Sending ε to zero, we obtain (14) with δ = s − 1 > 0.
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 [17].

Partial regularity -Proof of Theorem 2
Theorem 2 follows from, the higher integrability statement Theorem 1, the ε-regularity statement of Lemma 4 below and a well-known iteration argument.
With the higher integrability of Theorem 3 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.
where C * is chosen below. We consider the sequence of rescaled functions given by