Lipschitz bounds and nonautonomous integrals

We provide a general approach to Lipschitz regularity of solutions for a large class of vector-valued, nonautonomous variational problems exhibiting nonuniform ellipticity. The functionals considered here range amongst those with unbalanced polynomial growth conditions to those with fast, exponential type growth. The results obtained are sharp with respect to all the data considered and yield new, optimal regularity criteria even in the classical uniformly elliptic case. We give a classification of different types of nonuniform ellipticity, accordingly identifying suitable conditions to get regularity theorems.


Introduction
The aim of this paper is to provide a comprehensive treatment of Lipschitz regularity of solutions for a very large class of vector-valued nonautonomous variational problems, involving integral functionals of the type Here Ω ⊂ R n is an open subset with n ≥ 2 and N ≥ 1. In the following we shall assume the structure condition F (x, Dw) ≡F (x, |Dw|), which is natural in the vectorial case, where F : Ω × [0, ∞) → [0, ∞) is a suitably regular function (see Section 4.1 below for the precise assumptions). The vector field f : Ω → R N will be at least L n -integrable f ∈ L n (Ω; R N ) .
The notion of local minimizer used in this paper is quite standard in the literature. Definition 1. A map u ∈ W 1,1 loc (Ω; R N ) is a local minimizer of the functional F in (1.1) with f ∈ L n (Ω; R N ) if, for every open subsetΩ ⋐ Ω, we have F(u;Ω) < ∞ and F(u;Ω) ≤ F(w;Ω) holds for every competitor w ∈ u + W 1,1 0 (Ω; R N ). In the rest of the paper we shall abbreviate local minimizer simply by minimizer. We just remark that, thanks to (1.2) and Sobolev embedding, requiring that F(u;Ω) < ∞ for everỹ Ω in Definition 1 is the same to requiring that F (·, Du) ∈ L 1 loc (Ω). In this paper we deal with the following, classical Problem. Find minimal regularity assumptions on f and x → F (x, ·), guaranteeing local Lipschitz continuity of minima of the functional F in (1.1), provided this type of regularity holds when f ≡ 0 and no x-dependence occurs, i.e., F (x, z) ≡ F (z).
In particular, Du is locally bounded. We recall that f ∈ L(n, 1)(Ω; R N ) means that (1.6) f L(n,1)(Ω) := ∞ 0 |{x ∈ Ω : |f (x)| > λ}| 1/n dλ < ∞ , and also that L q ⊂ L(n, 1) ⊂ L n for every q > n. Moreover, denoting by ω(·) the modulus of continuity of c(·), the Dini continuity of c(·) amounts to require that The above theorem extends to general equations [46] and to systems depending on forms [65]; it also extends classical results of Uhlenbeck [69] and Uraltseva [70]; we again refer to Cianchi & Maz'ya [18,19,21] for global statements. The terminology is motivated by the fact that, for c(·) ≡ 1 and p = 2, this is another classical result of Stein [67]. It is optimal both with respect to condition (1.6), as shown by Cianchi [17], and with respect to (1.7), as shown by Jin, Maz'ya & Van Schaftingen [52]. The relevant fact here is that the conditions on f and c(·) implying local Lipschitz continuity are independent of p. In fact, when considering more general equations, they are independent of the vector field the divergence operator applies to; for this, see [46], and [1] for conditions (1.3)- (1.4).
In the case of nonuniformly elliptic operators, the problem of deriving sharp conditions with respect to data for Lipschitz regularity is considerably more difficult. This has been attacked only recently by L. Beck and the second author [3], but only for the case of autonomous functionals in the principal part, i.e., F (x, z) ≡ F (z). The outcome is that, when n > 2, condition (1.6) is still sufficient to guarantee the local Lipschitz regularity of minima, thereby revealing itself as a sort of universal property. In the case n = 2, the alternative, slightly stonger borderline condition L 2 (log L) a (Ω; R N ) with a > 2, implies Lipschitz continuity: (1.8) f ∈ L 2 (log L) a (Ω; R N ) ⇐⇒ Ω |f | 2 log a e + |f | dx < ∞ .
In this paper we deal with the general, fully nonautonomous case (1.1). This is by no means a technical extension as, in fact, when passing to the nonuniformly elliptic case, the role of coefficients drastically changes and they can no longer be treated via perturbation as in [47].
To give a glimpse of the situation, let us consider the so called double phase functional where H(x, z) :=H(x, |z|) := |z| p + a(x)|z| q , with 1 < p < q, 0 ≤ a(·) ∈ L ∞ (Ω). This functional has been originally introduced by Zhikov [72,73] in the setting of Homogeneization of strongly anisotropic materials, and the corresponding regularity theory has been studied at length starting by [2,19,20]. The functional in (1.10) changes its rate of ellipticity/coercivity -from p to q -around the zero set {a(x) = 0}. As shown in [37,39], already when f ≡ 0, local minima fail to be continuous if the ratio q/p is too far from 1, in depenence on the rate of Hölder continuity α. Specifically, the condition (1.11) q p ≤ 1 + α n , a(·) ∈ C 0,α (Ω) , α ∈ (0, 1] is necessary [37,39] and sufficient [2] to get gradient local continuity, thereby linking growth conditions of the integrand in the gradient variable, to the smoothness of coefficients. In particular, classical Schauder's theory generally fails. This is the main theme of this paper. Condition in (1.11) reveals a typical phenomenon occurring when nonuniform ellipticity is directly generated by the presence of the x-variable as in (1.10). In this case, it is indeed the very presence of x that makes functionals as in (1.10) fail to meet the standard, two-sided polynomial conditions with the same exponent, i.e., H(x, z) ≈ |z| p . We shall also deal with more drastic examples of such an interplay, as for instance where 0 < ν ≤ c 1 (·), c 2 (·) ≤ L. Here the dependence on x becomes even more delicate as it makes the ellipticity rate vary more drastically. Such integrands fail to satisfy the so-called △ 2 -condition, i.e.,F (x, 2t) F (x, t). This reflects in a loss of related integrability conditions on minimizers as one tries to use perturbation methods, that is, considering a specific point x 0 ∈ Ω and making small variations of x around x 0 . In other words, exp(c 2 (·)|Dw| p ) ∈ L 1 does not necessarily imply exp(c 2 (x 0 )|Dw| p ) ∈ L 1 and perturbation methods are therefore again banned. Exponential type functionals are classical in the Calculus of Variations starting by the work of Duc & Eells [36] and Marcellini [57]. In the nonautonomous version, they are treated for instance in the setting of weak KAM-theory, but only under very special assumptions and boundary conditions [38]. More recent progress is in [34], for f ≡ 0.
Nonuniform ellipticity is a very classical topic in partial differential equations, and it is often motivated by geometric and physical problems. Seminal papers on this subject are for instance [36,53,66,72,73]. In the setting of the Calculus of Variations there is a wide literature available, starting from the basic papers of Uraltseva & Urdaletova [71] and Marcellini [56][57][58][59]. More recently, the study of the nonautonomous case has intensified; many papers have been devoted to study specific structures as well as genereal non-uniformly elliptic problems [3-6, 8-11, 13, 30, 32-34, 50, 51]. Connections to related function spaces have been studied too [27,49,63].
The results obtained in this paper are very general and cover large classes of different models cases simultaneously. For this, a number of technical assumptions is necessary; see Section 4.5 below. Anyway, when applied to single models, such assumptions reveal to be minimal and produce sharp results. In the autonomous case F (x, z) ≡ F (z), they coincide with the sharp ones introduced in [3]. For this reason, and also to ease the reading, in this introductory part we shall present a few main corollaries of the general theorems, in connection to some relevant instances of nonuniformly elliptic functionals often considered in the literature. These models fall in three different general classes, detailed in Sections 1.1-1.3 below. We refer the reader to Section 2 for a full account of the notation used in this paper, while more remarks on nonuniform ellipticity are in Section 4.6 below.
Theorem 1 actually follows from Theorem 9 in Section 4.5 below and, as all the other results presented in this Introduction, comes along with explicit local a priori estimates. In particular, for splitting structures as

1.2.
Nonuniform ellipticity at fast rates and a different phenomenon. A prototype we have in mind is given by (1.12). Looking at the case of polynomial growth in Section 1.1, from (1.15) and (1. 16) we see that the required integrability rate of coefficients d increases with the ratio q/p. A naive, but seemingly natural bet, would then assert that the exponential case needs more stringent conditions on the integrability exponent d. On the contrary, the situation reverses, and any d > n implies local Lipschitz continuity: loc (Ω; R N ) be a minimizer of the functional in (1.12), such that c 1 (·), c 2 (·) ∈ W 1,d (Ω) with d > n and f satisfies (1.9). Then Du ∈ L ∞ loc (Ω; R N ×n ). The same applies to more general functionals, involving arbitrary compositions of exponentials, and therefore even faster growth conditions. Specifically, we fix sequences of exponent functions {p k (·)} and coefficients{c k (·)}, all defined on the open subset Ω ⊂ R n , such that We then inductively define, for every k ∈ N, the functions e k : Ω × [0, ∞) → R as and consider the variational integrals Functionals as in (1.22) have been studied at length in the literature also because they provide the best case study to test how far one can go in relaxing the standard uniform ellipticity assumptions; see [3,57] and related references. The nonautonomous case is of special interest as the sensitivity to the x-dependence is magnified by taking multiple compositions of exponentials; see comments after display (1.12). We have the following result, which, as also Theorem 4, is completely new already in the case f ≡ 0: Theorem 5. Let u ∈ W 1,1 loc (Ω; R N ) be a minimizer of the functional in (1.22) for some k ∈ N, under assumptions (1.20) and such that f satisfies (1.9). Then Du ∈ L ∞ loc (Ω; R N ×n ).
In other words, this fact brings functionals as in (1.12) closer to the realm of uniformly elliptic ones. The next step comes in fact in the subsequent section.
1.3. New results in the uniformly elliptic setting. New results follow in the classical uniformly elliptic setting too. This time the model is Theorem 6. Let u ∈ W 1,1 loc (Ω; R N ) be a minimizer of the functional in (1.23), under assumptions (1.4). If |f |, |Dc| ∈ X(Ω) as defined in (1.9), then Du ∈ L ∞ loc (Ω; R N ×n ). Moreover, there exists a positive radius R * ≡ R * (n, N, i a , s a , c(·)) ≤ 1 such that if B ⋐ Ω is a ball with r(B) ≤ R * , then holds for every s ∈ (0, 1), where c ≡ c(n, N, ν, L, i a , s a ).
In other words, f and Dc this time are required to have the same degree of regularity. Theorem 6 applies to (1.5) by taking A(x, t) ≡ c(x)t p /p and it is sufficient to require that Dc ∈ L(n, 1)(Ω) for n > 2. This is, already when f ≡ 0, a new regularity criterion, which goes beyond the known and classical one in (1.7). Indeed, Dc ∈ L(n, 1) implies that c(·) is continuous [67], but not necessarily with a modulus of continuity ω(·) satisfying (1.7). Moreover, this criterion works for the general cases as in (1.23), to which methods from [47] do not apply under the only considered structure assumption (1.4). When considered in the special case (1.5), it is i a = p − 2 and estimate (1.24) gives back the classical one valid for the p-Laplacean system in (1.5).
1.4. Calderón-Zygmund theory. In Theorems 1-6, we can replace (1.9) by the weaker f ∈ L n (Ω; R N ), getting, as a corresponding outcome, for instance that Du ∈ L p loc (Ω; R N ×n ) for every p ≥ 1; see Theorem 11 below. This result is new in the nonuniformly elliptic case and is in perfect accordance with the Nonlinear Calderón-Zygmund theory known for the uniformly elliptic one -see for instance [46]. For instance, considering the system in (1.5), Dc ∈ L n implies that c(·) ∈ VMO, the space of functions with vanishing mean oscillations [64]. At this point, Du ∈ L p loc (Ω), for every p ≥ 1, follows from the standard theory (see for instance [9,35]). In fact, we provide the first Calderón-Zygmund type estimates in problems with nonpolynomial growth conditions. An example is the following result, which is completely new already in the autonomous case: loc (Ω; R N ) be a minimizer of the functional in (1.22) for some k ∈ N, under assumptions (1.20) with n > 2, and such that f ∈ L n (Ω; R N ). Then e k (·, |Dw|) ∈ L p loc (Ω) for every p ≥ 1. 1.5. Obstacles. Applications follow to obstacle problems, leading to completely new and sharp results, already in classical, uniformly elliptic case. For instance, we give the first results for fast growth functionals as in (1.12), and these are new already in the case of smooth obstacles. For this we consider the functional where F is for instance one of the integrands considered in Theorems 1-6; here we of course consider the scalar case N = 1. Next we consider a measurable function ψ : Ω → R and the convex set K ψ (Ω) := {w ∈ W 1,1 loc (Ω) : w(x) ≥ ψ(x) for a.e. x ∈ Ω}. We then say that a function u ∈ W 1,1 We then have the following far reaching extension of classical theorems from [15,16,40,43,55]: loc (Ω) be a constrained local minimizer of F 0 in (1.25), where F : Ω× R n → R is one of the integrands from Theorems 1-5 with p ≥ 2 (whenever p is involved). Under the assumptions of such theorems with f ≡ 0, and assuming that ψ ∈ W 2,1 loc (Ω) with |D 2 ψ| ∈ X(Ω), it follows that Du ∈ L ∞ loc (Ω; R n ). This last result is new already in the classical p-Laplacean case F (x, z) ≡ |z| p /p, where it offers a criterion which is alternative to those given in [15,16] -see also [42] for double phase type functionals. In such papers Lipschitz estimates are obtained assuming that Dψ is locally Hölder continuous. Here we trade this last condition with |D 2 ψ| ∈ X(Ω), that in turn implies the mere continuity of Dψ. This is essentially the same phenomenon seen in Theorem 6, where the condition |Dc| ∈ X(Ω) replaces the Dini-continuity of c(·). We notice that in the constrained versions of Theorems 1 and 2 we can allow for p ≥ 1 provided µ > 0; for this see Remark 9 below. 1.6. Remarks and organization of the paper. Some of the methods here also extend to general scalar functionals, i.e., when minima and competitors are real valued functions. In this case there is no need to assume the radial structure F (x, Dw) ≡F (x, |Dw|). On the other hand, additional conditions ensuring the absence of the so-called Lavrentiev phenomenon are needed to build suitable approximation arguments, see for instance [33,37]. The radial structure is usually assumed in the vectorial case, otherwise singular minimizers might occur, even when data are smooth [62,68]. Again in the scalar case, we mention the recent, very interesting paper [50], where gradient regularity results are obtained for minimizers of functionals as in (1.25). These results cover functionals with polynomial growth and special structure -the double phase functional is an instance -under Hölder continuity assumptions on coefficients. Anyway, they miss to cover all the classes of integrands described in Sections 1.1-1.2.
The rest of the paper goes as follows. After Sections 2 and 3, containing notations and preliminaries, respectively, in Section 4 we describe in detail the assumptions and the main results of the paper, that is Theorems 9-12 in Section 4.5. These will imply, directly or with a few additional arguments, Theorems 1-8 above. We then proceed to Section 5, that contains the necessary approximation tools for the proofs. One word here: this is a delicate point, as the approximations considered must carefully match the shape of the a priori estimates found later, on one side, and reflect the original structure assumptions on the other. The core of the paper is Section 6, where we derive all the main a priori estimates. The proofs here involve a delicate version of Moser's iteration scheme. This is based on a peculiar choice of test functions suited to the structural assumptions considered. It goes via a finite step procedure taking advantage of suitable smallness conditions; by the way, this is sufficient to get the basic L p -estimates of Theorem 11. In turn, this is a preliminary ingredient used to make a nonlinear potential theoretic approach work; this last one is encoded in the abstract iteration Lemma 1 below. This approach works in the case n > 2, but breaks down in two dimensions n = 2, where more difficulties appear, as for instance already noticed in [3,[18][19][20][21]. In this case we take another path as devised in Section 6.7 below. We use a different interpolation type approach, eventually culminating in the use of Lemma 1 again. Sections 7 features the proofs of Theorems 9 and 11, combining the approximation scheme of Section 5 and the estimates in Section 6. Section 8 contains the results and the proofs for the uniformly elliptic case, i.e., the proof of Theorem 10. In Section 10 we demonstrate the derivation of Theorems 1-6. These are all direct corollaries of the main results, but Theorem 3. This in fact requires some additional arguments to reach the = borderline case in the first bound from (1.19), thereby reconnecting to the known literature [2,19,20] in the case f ≡ 0. Section 11 contains applications to obstacle problems. Finally, Section 12 features some auxiliary technical facts aimed at making certain computations in Sections 6 and 8 legal.

Basic notation
In the following Ω ⊂ R n denotes an open domain, and n ≥ 2 and there is no loss of generality, in assuming that Ω is also bounded, as all our results are local in nature. We denote by c a general constant larger than 1. Different occurrences from line to line will be still denoted by c. Special occurrences will be denoted by c * ,c or likewise. Relevant dependencies on parameters will be as usual emphasized by putting them in parentheses. We denote by B r (x 0 ) := {x ∈ R n : |x − x 0 | < r} the open ball with center x 0 and radius r > 0; we omit denoting the center when it is not necessary, i.e., B ≡ B r ≡ B r (x 0 ); this especially happens when various balls in the same context will share the same center. Finally, with B being a given ball with radius r and γ being a positive number, we denote by γB the concentric ball with radius γr and by B/γ ≡ (1/γ)B. In denoting several function spaces like L p (Ω), W 1,p (Ω), we shall denote the vector valued version by L p (Ω; R k ), W 1,p (Ω; R k ) in the case the maps considered take values in R k , k ∈ N. We shall often abbreviate L p (Ω; R k ) ≡ L p (Ω), W 1,p (Ω; R k ) ≡ W 1,p (Ω). We denote {e α } N α=1 and {e i } n i=1 standard bases for R N and R n , respectively; we shall always assume n ≥ 2 and N ≥ 1. The general second-order tensor of size (N, n) as ζ = ζ α i e α ⊗ e i is identified with an element of R N ×n . The Frobenius product of second-order tensors z and ξ is defined as z · ξ = z α i ξ α i so that ξ · ξ = |ξ| 2 , and this is the norm we use here for tensors, vectors, matrixes. For instance, if v, u ∈ R k , then v · u = v i u i . The gradient of a map u = u α e α is thus defined as Du = ∂ xi u α e α ⊗ e i ≡ D i u α e α ⊗ e i , and the divergence of a tensor ζ = ζ α i e α ⊗ e i as div ζ = ∂ xi ζ α i e α . When dealing with the integrands of the type F : R N ×n → [0, ∞), as the one considered in (1.1), second differential of ∂ zz F (z) , whenever z ∈ R N ×n . For the rest of the paper we shall keep the following notation: With B ⊂ R n being a measurable subset with bounded positive measure 0 < |B| < ∞, and with g : B → R k , k ≥ 1, being a measurable map, we denote Finally, in the following we denote The actual value of 2 * when n = 2 will be clear from the context.

Potentials, functions spaces, iteration lemmas
With g ∈ L 2 (B r (x 0 ); R k ) and B r (x 0 ) ⊂ R n being any ball, we consider the following nonlinear potential, that will play a crucial role in this paper: This quantity naturally relates to the standard, truncated Riesz potential in the sense that As a matter of fact, P g 1 can be used as an effective replacement of the original Riesz potential when dealing with nonlinear problems. Actually, its mapping properties coincide with those of the classical Riesz potentials on those function spaces that are in a sense smaller than L 2 . We refer to [26,46,61] for more information on such nonlinear potentials and for recent results on Nonlinear Potential Theory in the setting of this paper. The space X(·) in (1.9) plays a special role to ensure the local boundedness P 1 . Indeed, given concentric balls B R ⊂ B R+r ⊂ R n , with r > 0 and R + r ≤ 2, and g ∈ L 2 (B R+r ; R k ), the following inequalities hold: where in the last of the two inequalities c(a) → ∞ when a ց 2. For this we refer to [3,46]. Notice that in the right-hand side of (3.1) 2 we find the Luxemburg norm of of the Orlicz space L 2 (log L) a in (1.8), which is defined as The two following inequalities are well-known and hold for any τ > 0: (3.2) τ g L(n,1)(BR+r) ≤ c(n) g L n+τ (BR+r) and τ g L 2 (log L) a (BR+r) ≤ c(a) g L 2+τ (BR+r ) .
We conclude with another, more classical iteration lemma (see references of [3]).

Assumptions and general results
In this section we are going to describe a number of conditions on the integrand F in (1.1) implying our main results, that is Theorems 9-12 in Section 4.5 below; in turn, these will imply Theorems 1-8 from the Introduction. Remark 1. When considering F , its partial derivatives involving the x-variable will be assumed to be at least Carathéodory regular; those not involving the x-variable will be assumed to be continuous. For those functions being genuinely Carathéodory regular -essentially those involving the x-variable and the function g 3 below -when prescribing a pointwise condition, for instance on ∂ xF ′ (x, t) we shall say that this holds for (x, t) ∈ Ω × [T, ∞) or the like, meaning that, for each t ≥ T , the corresponding condition is satisfied by ∂ x F ′ (x, t) for a.e.
4.1. Basic structural assumptions, and consequences. We assume that F has radial structure, i.e., there existsF : Now we describe the minimal and standard assumptions qualifying the functional in (1.1) as elliptic. For this, we use three locally bounded functions g i : Ω × (0, ∞) → [0, ∞), for i ∈ {1, 2, 3}. The first two g 1 , g 2 are continuous and serve to bound the lowest and the largest eigenvalues of ∂ zz F , respectively. The third one g 3 is Carathéodory regular and controls the growth of mixed derivatives. Specifically, we assume that there exists T > 0 such that where 0 ≤ h(·) ∈ L n (Ω) and we assume also that inf x∈Ω g 1 (x, T ) > 0. Needless to say, all the maps and functions considered in (4.1)-(4.2) are at least Carathéodory regular. Using as in (1.3) the notationã(x, t) :=F ′ (x, t)/t, defined for (x, t) ∈ Ω × (0, ∞), we assume, for fixed numbers γ > 1, µ ∈ [0, 1], and for every x ∈ Ω, the minimal γ-superlinear growth of the lowest eigenvalue of ∂ zz F (x, ·), that is are non-decreasing on (0, ∞) .
We finally assume that the ratio g 2 /g 1 is almost non-decreasing with respect to t, i.e., holds for all x ∈ Ω and some c a ≥ 1.

Remark 2.
In most of the relevant models it turns out to beã(·) ≡ g 1 (·), and this justifies the double assumption in (4.3). Assumptions in (4.2) are bound the quantify ellipticity of ∂ zz F (·, z) only outside the ball {|z| < T } (recall we are assuming ν > 0), and this allows to cover functionals loosing their ellipticity properties on a bounded set. This condition not only adds more generality, but helps simplifying the treatment in the case of nonuniformly elliptic problems. We could do the same also with respect to the partial derivative ∂ xz F in (4.2) 4 , but we prefer not to follow this path as this would only add useless technical difficulties, while not covering more examples.

4.2.
The energy functions. These are two functions bound to quantify the minimal energy controlled by the functional in (1.1); they will play a crucial role in the rest of the paper. For every (x, t) ∈ Ω × (0, ∞), we set 4.3. Quantifying nonuniform ellipticity. Here we quantify ellipticity and growth of F in the nonuniformly elliptic case. For this, we consider numbers d, σ,σ ≥ 0 such that Here it is ϑ ≡ ϑ(γ) = 1 when γ ≥ 2, and ϑ = 2 otherwise. We assume that x → g 1 (x, t) ∈ W 1,d (Ω) for all t ≥ T and that ∂ x g 1 (·) is Carathéodory regular on Ω × [T, ∞). Then, for every (x, t) ∈ Ω × [T, ∞), and for a fixed constant c b ≥ 1, we consider the assumptions Comments on the meaning of (4.14)-(4.16) can be found in Section 4.6 below.

4.4.
The uniformly elliptic case. Here we instead describe conditions relevant to the uniformly elliptic case, and therefore to models as in (1.23). Such conditions are a slightly more general version of those considered in [18,19,54], that are classical. We retain the structure conditions (4.1)-(4.3) from Section 4.1 but, instead of using (4.12)-(4.16) from Section 4.3, for a fixed constant c u ≥ 1 this time we consider and these are going to be used in the nonuniformly elliptic setting in the uniformly elliptic one, respectively. Accordingly, we also gather the parameters influencing the constants in the a priori estimates as data := (n, N, ν, γ, T, c a , c b , d, σ,σ, a) , data u := (n, N, ν, γ, T, c u , a) .
The main Lipschitz regularity result in the uniformly elliptic case is in the next Theorem 10. Let u ∈ W 1,1 loc (Ω; R N ) be a minimizer of F in (1.1), under assumptions set u in (4.18), and with f, h ∈ X(Ω) as defined in (1.9). Then Du ∈ L ∞ loc (Ω; R N ×n ). Moreover, there exists a positive radius R * ≡ R * (data u , h(·)) ≤ 1 such that if B ⋐ Ω is a ball with r(B) ≤ R * , then the intrinsic estimate holds for every s ∈ (0, 1), where c ≡ c(data u ).
Remark 5. In Theorems 9-11, as well as in the other estimates in this paper, the constants depending on data and data u blow-up when T → ∞ (complete loss of ellipticity), and remain bounded when T → 0 (full recovery of ellipticity), as long as the quantity ν ≡ ν(T ) in (4.6) stays bounded away from zero. This is for instance the case in Theorems 1-5; see Section 10 below. As a matter of fact, there is an additional dependence of the constants on the specific operator considered. Indeed, the dependence on T typically shows up via quantities controlled by ã(·, T ) L ∞ (T 2 + µ 2 ); this can be in turn controlled as ã(·, 1) L ∞ (T γ + µ γ ) for T ≤ 1 via (4.3). All in all, the dependence of the constants on the specific operator (i.e., ∂ x F ) appears only through ν defined in (4.6), and ã(·, 1) L ∞ .
We give another general result follows taking into account an intermediate form of nonuniform ellipticity. This is for instance tailored to the case of functionals with special structure as the double phase one in (1.10). More comments on this will follow in Section 4.6 below.
Theorem 12. Let u ∈ W 1,1 loc (Ω; R N ) be a minimizer of F in (1.1), under assumptions set in (4.18), with f as in (1.9). Moreover, replace (4.13) and (4.16) by respectively. Then Du ∈ L ∞ loc (Ω; R N ×n ) and (4.20) holds. Remark 6. We notice that replacing the integrand , does not change the set of minimizers, and gets another functional still satisfying the conditions of Theorems 9-12. Moreover, asF (x, 0) ≥ 0, once the estimates in (4.20)-(4.22) are proved in the case F 0 (x, z) is considered, then they also hold in the original case. Therefore for the rest of the paper we can always assume thatF (x, 0) = 0 and it followsF 4.6. Different measures of nonuniform ellipticity. What do we call nonuniform ellipticity of the integrand F here? To provide a measure, in the autonomous case F (x, z) ≡ F (z), it is rather standard to use the ellipticity ratio The occurrence of nonuniform ellipticity then refers to the case when R F (z) → ∞ as |z| → ∞; accordingly, the uniformly elliptic case occurs when R F (·) ≈ 1. In the nonautonomous case this leads to define the same pointwise quantity R F (x, z), by taking into account the xdependence in the right-hand side of (4.25). In this paper we have considered assumptions aimed at bounding R F (x, z) pointwise, i.e., On the other hand, the ratio R F (x, z) misses to properly encode the full information in the nonautonomous one. This leads to consider the non-local quantity with B ⊂ Ω being any ball. The new ratio R F (z, B) naturally occurs in integral estimates and, in fact, turns out to be the right quantity to describe nonuniform ellipticity in the nonautonomous case. Obviously, from the definitions it follows that , with equality in the autonomous case.
Remark 7 (Double face of the double phase functional (1.10)). To further motivate (4.27), recall that for general uniformly elliptic problems one typically recovers classical theories as Schauder's [54] and Calderón-Zygmund's [12]. This is in general not the case for the double phase integral in (1.10), due to the counterexamples in [37,39]. Nevertheless the integrand H(·) turns out to be uniformly elliptic when using the quantity in (4.25) as test, in the sense that R H (·) ≈ 1. On the other hand, when considering a ball B such that In such sense, the quantity in (4.27) appears to be more for nonautonomous problems, thereby resolving the above ambiguity. In this case, the assumptions (1.19) provide a way to correct the growth of R H (z, B) with respect to |z| with the smallness of |B|, as pointed out in the Introduction.
In this paper we consider different assumptions, playing with the parameters σ,σ in (4.14)-(4.16), in order to tune different degrees of nonuniform ellipticity, involving both R F (·) and R F (·). In Theorems 9-10, we prescribe a direct pointwise bound on R F (·) as in (4.26), and then control the behaviour with respect to x of the derivatives of F via (4.14)-(4.15). This has the overall effect of providing an indirect control on R F (·) too. In Theorem 12, we bound R F (·) 1, that is, uniform ellipticity is assumed in the pointwise sense, still allowing for nonuniform ellipticity in the nonlocal sense of (4.27). This leads for instance to get better bounds and opens the way to Theorem 3, dealing with the intermediate case of double phase functionals in the sense of Remark 7.

Approximation of integrands
Here we implement a truncation scheme aimed at approximating the original integrand F in (1.1) with a family {F ε } of integrands with standard polynomial growth, and converging to F locally uniformly. The new integrands preserve structure properties as (4.1)-(4.2), with corresponding control functions g i,ε , still satisfying relations as those of the original ones. For this we shall a few arguments used in [3] as a starting point. In this section we shall permanently assume that (4.1)-(4.4) are in force, so that their consequences (4.5)-(4.10) can be used as well. Additional assumptions as (4.14)-(4.16) shall also be considered; in case, this will be explicitly mentioned. In the following we use a parameter ε that will always be such that 0 < ε < min{1, T }/4.

General setup. Given T and µ in (4.2) and (4.3), respecively, we introduce the numbers
for every x ∈ Ω, where γ also appears in (4.3); accordingly, we define and, finallyā In view of (4.1) and (4.24), it follows is strictly convex and non-decreasing for all x ∈ Ω F ε → F uniformly on compact subsets ofΩ × R N ×n as ε → 0 . The above definitions lead to introduce new control functions g i,ε : Ω × [0, ∞) → (0, ∞), i ∈ {1, 2, 3} as: where the constants g 1 and g 2 are defined by We next introduce also the truncated counterparts of the functions defined in (4.11), i.e., In the following, recalling (5.1), we shall repeatedly use that

Lemma 4. Under assumptions (4.1)-(4.4), and for every
In particular, it follows that the function G ε in (5.10) is convex. • If (4.16) is also in force for some σ ≥ 0, then it holds that In particular, if g 2 /g 2 ≤ c b holds as in (4.23), then it also holds that • If (4.14) is also in force, then, with g 1 as in (5.9) it holds that • Again for all (x, t) ∈ Ω × [T, ∞), the following holds: Proof. The second property in (5.23) readily follows from (4.5) and the definitions in (5.6)-(5.7), also using the fact that t → (t 2 + µ 2 ε ) (γ−2)/2 t is increasing. As for the first one, it again follows from (4.4) once we notice that (4.6) implies and that the definitions in (5.6)-(5.7) imply As for (5.24), notice that, for T ≤ t ≤ T ε we have The proof of (5.26) is a straightforward consequence of the definition in (5.6) and (4.14) when t ≤ T ε . In the case t > T ε , we instead have, also using (4.14) that is, (5.26). Let us now take care of (5.27) 1 . For t ∈ [T, T ε ), using (4.6) and (5.11), we see that (recall that ε ≤ T /4) while, when t ≥ T ε by analogous means, we havē .
, and therefore we also have that G ε1 (·, |Dw|) ∈ L p (B). Next, we denote ε (x, s)s ds and in the following we always take x ∈ B. In the case it is |z| ≤ ε 2 , by (4.3) we easily have When ε 2 ≤ |z| ≤ T ε2 , recalling thatã ε1 (x, |z|) ≡ã ε2 (x, |z|), by also using the information in the last display we find Finally, notice that (4.3) and (5.11) therefore, when |z| > T ε2 , using also the last inequality in (5.40), we get Using the content of the last fours displays, and using also Hölder's inequality, we get for c ≡ c(ν, γ, p). On the other hand, observe that Combining the previous two displays yields where c ≡ c(n, N, ν, γ, c a , c b ). By using again (5.27) 1 and Hölder's inequality, we get Combining the content of the last two displays and recalling (5.3) and (5.40) we arrive at (5.38). As for (5.39), this follows from (5.5) 5 and (5.38) letting ε 1 → 0. Indeed, notice that Fatou's lemma works for the left-hand one; as for the right-hand side, we use again that G ε (x, t) G(x, t) and the second information in (5.29), and finally Lebesgue dominated convergence.

A priori estimates
In this section we develop basic a priori estimates. These are obtained for solutions to certain elliptic systems associated to the integrands defined in Section 5, (5.3). Unless differently specified, we shall permanently assume that set in (4.18) is in force; all properties (4.1)-(4.16) will be therefore available as well. With 0 < ε < min{1, T }/4 and B ⋐ Ω being a ball such that r(B) ≤ 1, we consider a weak solution u ∈ W 1,γ (B; R N ) to the system with a ε : Ω × R N ×n → R N ×n being defined as in (5.4). This setting will be kept for the rest of Section 6. In particular, all the balls considered in the following will have radius ≤ 1; eventually, we shall consider additional restrictions of the type r(B) ≤ R * , where R * will be a (small) threshold radius to be determined as a function of the fixed parameters of the problem. From (6.1) it follows that u is a minimizer of the functional Finally, the functions t →ā ε (·, t) and g 1,ε (·, t) are non-decreasing when γ ≥ 2; this is indeed an easy consequence of assumption (4.3) 1 . A direct consequence of (5.14)-(5.15) is 3) trivially follows from (5.15) 2 ; otherwise (5.15) 1 , and then (5.14), give |ā ′ . Similarly to (5.14), by (5.13), we have that holds this time for all (x, z) ∈ Ω × R N ×n (recall that z →ā ε (x, |z|) is continuous). From (5.12)-(5.13) we can apply Proposition 6 from the Section 12, and this yields In turn, this implies that . This follows for instance from the results in [28,Theorem 1.5], together with the expression of Da ε (·, Du). For this, we remark that the crucial point to apply the results from [28] is that the set of non-differentiable points of the partial map t →ā ε (x, t) is {ε, T ε } and is therefore independent of x (here this holds for every x, but an a.e. condition is still allowed by [28]). Specifically, with slight abuse of notation, let us write and here we are denoting by 1 D (·) the indicator function of the set D := R \ {ε, T ε }. Then, recalling that |Du| ∈ W 1,2 loc (Ω) by (6.5)-(6.6), we have Notice that, exactly as in the usual autonomous case, the presence of 1 D (|Du|) in (6.7)-(6.8) accounts for the fact that terms asā ′ ε (x, |Du|)D s |Du| are interpreted as zero at those points where |Du| ∈ {ε, T ε }, i.e., whereā ′ ε (x, |Du|) alone does not make sense; see [28]. These same arguments apply to G ε too; indeed, notice that for every x ∈ Ω, the function t → G ε (x, t) is differentiable at every point but T and by (4.14) satisfies the assumptions of [28,Theorem 1.5]. So, it is G ε (·, |Du|) ∈ W 1,2 loc (B) and, on {|Du| > T }, we have for every i ∈ {1, . . . , n}, where we have also used that holds for every x ∈ Ω.
Proof. Indeed, (6.11) is trivial by (5.14) ifā ′ ε (x, |z|) ≥ 0. Otherwise, we can estimate simplȳ a ′ ε (x, |z|)|λ · z α | 2 ≥ā ′ ε (x, |z|)|z α | 2 |λ| 2 for every α and then use (5.15) 1 . 6.1. Caccioppoli inequality for powers, when n > 2. Up to Section 6.3 included, we concentrate on the case n > 2, and for the following we set The main result here is be a solution to (6.1), under assumptions set in (4.18) for n > 2, and replace (4.13) 1 by Then, for each p ∈ [1, ∞), there exists a positive radius Needless to say, (6.13) is implied by (4.13) 1 for s * = 1. The proof of Proposition 1 will take this and the subsequent Sections 6.2-6.3; in the following, all the balls considered but B, will be concentric with B ς ⋐ B ̺ from the statement. We recall it is r(B) ≤ 1; the size of R * will be chosen in due course of the proof. Observe that all the foregoing computations, except those involving f , still work in the case n = 2; this will be treated in Section 6.7 below. To start with the proof of Proposition 1, by (6.5)-(6.6) we pass to the differentiated form of system (6.1), that is which holds for all ϕ ∈ W 1,2 0 (B; R N ). We now consider concentric balls B ς ⊂ B τ1 ⋐ B τ2 ⊂ B ̺ ; in particular it is τ 1 < τ 2 ≤ ̺ ≤ 1. By (6.5), the integral identity in (6.15) can be tested against
Notice that s * /n − s * /d < χ − 1 if and only if s * < 2n/(n − 2), and therefore the second condition in (6.13) implies, again together with the first one, that This, in turn, implies that {κ i } and {s i } are increasing sequences; moreover, it holds that Notice that this implies Again (6.27) implies (6.29) so that from the first relation in (6.28) it also follows that κ i → ∞. For 0 < ς ≤ τ 1 < τ 2 ≤ ̺, we consider a sequence {B ̺i } of shrinking, concentric balls, (6.25), elementary manipulations also based on (6.26) and (6.28) give that , so that (6.30) (recall that 2 * = 2mχ by (6.12)) becomes Iterating the above inequality yields that Now, from (6.29) we deduce that The function t → t/χ t is decreasing on [1/ log χ, ∞) and using this fact one sees that We then write and notice that, for every integer i ≥ 1, we have where c ≡ c(n, d, σ,σ) in all cases. Using the content of the last three displays yields where c depends on n, d, σ,σ and h L d (B) , and it is β ≡ β(n, d, σ,σ). Notice that such constants blow-up when χ → 1 + σ +σ; in particular, this happens when d → n. Using (6.34) in (6.31), and keeping (6.32) in mind, yields ) and β ≡ β(n, d, σ,σ), for every i ∈ N such that κ i ≤κ.
6.3. Sobolev regularity. By (6.13) we have 1 ≤ s * < s 1 = 2m(1 + σ +σ) so that, for every integer index i ≥ 1, we consider the interpolation inequality Let us show that, thanks to (6.13), there existθ ≡θ(n, d, σ,σ) < 1 and i 1 ∈ N such that Indeed, for this it is sufficient to observe that Note that the last inequality is actually equivalent to (6.13) and that (6.13) in fact reduces to (4.13) for s * = 1. Now, we consider the number p > 1 for the statement of Proposition 1, and determine another index k ≡ k(p) > i 1 such that s k+1 ≥ p; accordingly, we consider the number κ k related to s k via (6.28). We now choose the numberκ ≡κ(p) in (6.24) asκ := κ k , and accordingly we determine R * ≡ R * (p) via (6.24). It follows that (6.35) holds in the case i ≡ k and therefore we can plug (6.36) in it, thereby obtaining On the other hand, as k > i 1 , then (6.38) holds with i ≡ k; we can therefore apply Young inequality in (6.39); this yields Lemma 2 with the choice Z(t) ≡ G ε (·, |Du|) L s k+1 (Bt) , finally renders that ). This last inequality holds provided the bound in (6.13) holds. All in all, recalling (5.16) 2 , (6.22) and that p ≤ s k+1 , completes the proof of Proposition 1 with β p := β k(p) , θ p := θ k(p) . We remark that in (6.14) the exponents θ p , β p can be replaced by exponentsβ,θ ≡β,θ(n, d, σ,σ) that are independent of p. For this, observe that when k, p → ∞. The only dependence on p in (6.14) comes through the threshold radius R * ; it is R * (p) → 0 as p → ∞ unless f ≡ 0.

6.4.
A Lipschitz bound in the homogeneous case f ≡ 0. The above reasoning, eventually culminating in Proposition 1, immediately leads to a priori Lipschitz estimates when f ≡ 0. The result, when combined with the approximation of Section 7 below, extends those in [57,58,60] to the case of nonautonomous functionals with superlinear growth as in (1.1). For this, the key observation is that it is not necessary to consider balls with small radii R * as in (6.24), as the last term in (6.23) does not appear. Therefore we can take everywhere, and in particular in (6.14), R * = 1, independently of the value of p. It follows we can let p → ∞ in (6.14), and recalling (6.41) we conclude with Proposition 2. Let u ∈ W 1,γ (B; R N ) be a solution to (6.1), under assumptions set in (4.18) for n ≥ 2 and with f ≡ 0. Moreover, replace (4.13) by the weaker σ +σ < 1/n − 1/d. If B ς ⋐ B ̺ are concentric balls contained in B, then holds with c ≡ c(data, h L d (B) ) ≥ 1,β,θ ≡β,θ(n, d, σ,σ) > 0.
Notice that here we are using Proposition 1 with the choice s = 1. Notice also that Proposition 1 refers to the case n > 2. The remaining two dimensional case can be obtained via minor modifications to the proof of Proposition 1, by choosing 2 * /2 large enough (see (2.2)) in order to get χ > 1 in (6.12). Anyway, the two dimensional case n = 2 will be treated in Section 6.7 directly for the general case f = 0. In that situation the proof cannot be readapted from the one of Proposition 1 as for Proposition 2.
Case 2: γ ≥ 2 in (4.3), and therefore it is ϑ = 1. In this case we use that the function t →ā ε (·, t) is non-decreasing, so thatā ′ ε (·) is non-negative (when it exists). We notice that In turn, we have For (VI) z , we again use (6.7) andā ′ ε (·) ≥ 0 ti estimate via Young inequality as follows: As for (V) z,2 , by letting (6.50) we have, again using thatā ′ ε (·) ≥ 0 On the other hand, we have Assembling the content of displays from (6.48) to (6.51), we again conclude with (6.47), but this time with ϑ = 1. We proceed estimating the x-terms coming from (6.47), using (5.12) 3 , (5.32) as follows: with c ≡ c(data) and arbitraryε ∈ (0, 1). Finally, the estimate of the terms involving f can be done by using Young's inequality and (5.35) as follows: for c ≡ c(data) and arbitraryε ∈ (0, 1). Collecting the estimates in the last three displays to (6.47), recalling that recalling thatḠ ε (·) ≥ 1, and selectingε > 0 sufficiently small in order to reabsorb terms, we complete the proof of Lemma 8. 6.6. Nonlinear iterations. In this section we finally derive pointwise gradient bounds. This goes via Lemma 9 and Proposition 3 below.
Lemma 9. Let u ∈ W 1,γ (B; R N ) be a solution to (6.1), under assumptions set in (4.18) for n > 2. If B 2r0 (x 0 ) ⋐ B is a ball such that x 0 is a Lebesgue point of both |Du| and h(·), then Proof. Notice that we can assume that |Du(x 0 )| > T , otherwise (6.53) is trivial. Let us first notice that x 0 is also a Lebesgue point of x → G ε (x, |Du(x)|) and it is i.e., the right-hand side denotes the precise representative of G ε (·, |Du(·)|) at the point x 0 . Indeed, notice that As x 0 is a Lebesgue point for Du, t → G ε (x 0 , t) is locally Lipschitz-regular, and Du is locally bounded, we have As for the term C 1 (·), we have where c ≡ c(ν, γ, Du L ∞ (Br (x0)) ). Therefore, being x 0 a Lebesgue point of h(·), we infer C 1 (r) → 0 as r → 0. This fact, together with (6.56)-(6.57), yields (6.55). Thanks to (6.42) we can verify ( . Applying Lemma 1, inequality (3.4) yields (6.53).
If B ς ⋐ B ̺ are concentric balls contained in B, then, for every p ≥ 1, there holds Proof. We can confine ourselves to prove (6.69) for sufficiently large p, and we consider Notice that such a choice is possible thanks to (6.68), making the denominator of the last quantity different than zero. In the following lines all the balls will be concentric to B ̺ . We look back at the proof of Proposition 1, take κ = 0 in the test function ϕ s = η 2 G ε (x, |Du|)D s u for s ∈ {1, . . . , n}, and perform exactly the same calculations made there up to (6.19). For the terms (I) f -(III) f involving the right-hand side f , we notice that as κ = 0 the test functions ϕ s used in the proof of Propositions 1 and Lemma 8 do coincide. Therefore we can use estimate (6.52), where c ≡ c(data) andε ∈ (0, 1); here S 3 and S 4 have been defined in (6.44) and (6.45), respectively. All together, choosingε > 0 small enough and re-absorbing terms in a standard way, we obtain for c ≡ c(data). As it is |Dη| 1/(τ 2 − τ 1 ), elementary manipulations on (6.71) give so that Sobolev embedding gives using the interpolation inequality Using (6.68) and (6.70), we see that λ p (1 + σ +σ) < 1, thus Young inequality gives with c ≡ c(data, h L d (B) ) and Lemma 2 with the choice Z(t) ≡ G ε (·, |Du|) L p (Bt) now gives (6.69).
We finally come to the a priori gradient bound in the two dimensional case.

Proofs of Theorems 9 and 11
We start with the proof of Theorem 11, where, in particular, we assume f ∈ L n (Ω; R N ) and n > 2. We fix p as in statement and, without loss of generality, we assume that p > 1 + σ (σ being as in (4.13)). Then, for every integer j ≥ 1, Next, we determine R * ≡ R * (data, f (·), p) ≤ 1 according to Proposition 1. Pay attention here; with some abuse of notation, the f used here is not the same from Proposition 1, but rather corresponds to f from (6.1) in the context of Proposition 1 (and thanks to (7.1), f j corresponds to f in Proposition 1). Accordingly, we fix a ball B ⋐ Ω such that r(B) ≤ R * . We consider a decreasing sequence of positive numbers {ε j } such that ε j ≤ min{1, T }/4 for every j ∈ N, and, accordingly, we consider the families of fuinctions . Notice now that any minimizer u of the functional F in (1.1) belongs to W 1,γ loc (Ω; R N ) by (5.17) 1 . This allows to define u j ∈ u + W 1,γ 0 (B; R N ) as the solution to for every j ≥ 1, where c ≡ c(n, N, ν, γ, T ). This implies we can assume that, up to a not relabelled subsequence, Du j ⇀ Dũ weakly in L γ (B; R N ×n ) and u j →ũ strongly in L n n−1 (B; R N ), for someũ ∈ u + W 1,γ 0 (B; R N ). Notice that Proposition 1 applies to u j with the choice s * = 1, as (6.13) is satisfied assuming that σ +σ < 1/n − 1/d, which is the case here by (4.13) in set. The application of Proposition 1, and (6.14), now give that holds for every s ∈ (0, 1), for new exponents β p , θ p ≡ β p , θ p (n, γ, d, σ,σ, p) > 0, where we have also used (7.2) to bound the right-side coming from (6.14); c depends on data and h L d (B) . We fix j 0 ∈ N, and apply (5.29) with j ≥ j 0 in (7.3), to get Letting j → ∞ in the above inequality, and using weak lower semicontinuity (recall that G j0 (·) is convex by (5.23), ε ≡ ε j0 ), yields This holds for every j 0 ∈ N and therefore, by finally letting j 0 → ∞ (by Fatou's lemma) and recalling (5.29), we conclude with for every s ∈ (0, 1), where c ≡ c(data, h L d (B) ) and β p , θ p ≡ β p , θ p (n, γ, d, σ,σ, p) > 0. Next, we trivially write whenever s ∈ (0, 1). Properties (7.1)-(7.2), Hölder and Sobolev-Poincaré inequalities give where c is independent of s, j and we have used minimality of u j . Using (5.38) with ε 1 ≡ ε j , ε 2 ≡ ε j0 , w ≡ u j , B ≡ sB (recall it is p > 1 + σ) and (7.3), we have where c is independent of s, j, j 0 . Again Sobolev-Poincaré inequality (7.1) and (7.2) give , with c independent of s, j, j 0 . Using this last inequality and (7.7)-(7.8) in (7.6), and finally letting j → ∞, lower semicontinuity yields Notice that we have used (5.20) and Lebesgue's dominated convergence theorem to get F j (u; B) → F(u; B). In turn, again notice that (7.5) ensures that G(·, |Dũ|) ∈ L p (sB) and therefore allows to apply (5.39); this yields F (·, Using this information in (7.5) yields (4.22) and concludes the proof of Theorem 11. Observe that, in order to justify the content of Remark 4 it is sufficient to notice that making the a priori estimate (7.3) only requires the bound σ +σ < 1/n − 1/d.
We now come to the proof of Theorem 9. We can again use the same approximation employed for Theorem 11. Using this time estimate (6.58) for the case n > 2 and estimate (6.74) when n = 2, together with (7.2), we find for every s ∈ (0, 1), where c, β, θ have the same dependencies as in (6.58). It follows that for every s ∈ (0, 1) there exists M s such that Du j L ∞ (sB) ≤ M s , for every j ∈ N. Using a standard diagonalization argument we infer that, up to a not relabelled subsequences, we have that u j ⇀ * ũ in W 1,∞ loc (B; R N ) forũ ∈ u+W 1,γ 0 (B; R N ) and Dũ L ∞ (sB) ≤ M s . Moreover, we can repeat verbatim the argument of Theorem 11 leading to (7.10). Now, denoting j s the first integer such that 1/ε js > M s , from the very definition of G j ≡ G εj in (5.10), it follows that G j (·, |Du j |) L ∞ (sB) = g 1 G(·, |Du j |) L ∞ (sB) , for every j ≥ j s . Therefore, letting j → ∞ in (7.11), and using (7.10), yields (4.20) and the proof of Theorem 9 is complete too.
Remark 8. Let us discuss the case f ≡ 0. We start by the case n > 2; the first relation in (6.64) is already implied by σ +σ < 1/n − 1/d. Moreover, notice that the second condition in (6.64) appears only when f ≡ 0. This is the only point in the proof of Theorem 9 where the full bound in (4.13) 1 is required; otherwise σ +σ < 1/n − 1/d is sufficient. Alternatively, one can apply directly Proposition 2 instead of Proposition 1 in the proof of Theorem 9. When n = 2 and f ≡ 0, (6.76) turns into (1 + δ)ϑσ/2 < 1 and δϑσ/2 < 1, that are implied by σ +σ < 1/2 − 1/d by taking δ small enough (see estimate of T 4 in Proposition 4). Finally, notice that in the case γ ≥ 2, the minimum in (4.13) is attained by 1/n − 1/d. All in all, we have justified the content of Remark 3.

Uniform ellipticity and proof of Theorem 10
Here we work under assumptions set u in (4.18); we keep the full notation introduced in Sections 5 and 6. In particular, we keep on considering a solution u ∈ W 1,γ (B; R N ) to (6.1) in a ball B ⋐ Ω such that r(B) ≤ 1. With the current choice of g 1 , g 2 , g 3 as in (4.17), we apply the constructions laid down in Section 5, thereby obtaining, in particular, the new functionsã ε , g 1,ε , g 2,ε , g 3,ε in (5.2) and (5.6)- (5.8). Notice that the last three functions are now independent of x, as well as G ε defined in (5.10). Notice that (4.17) 2 from set u implies the validity of (4.4) with c a ≡ c u , and the validity of (4.16) with c b ≡ c u and σ = 0. Therefore we can use all the properties from Section 5, and displayed through Lemmas 3-6, implied by assumptions (4.1)-(4.4) and (4.16). In particular, we can use Lemma 6 with σ = 0. In addition to such properties, we have Lemma 11. Under assumptions set u in (4.18) • The following inequalities hold for every t ∈ [T, ∞): • Moreover, for every x ∈ Ω and t ≥ 0, it holds that Proof. Properties (8.1) are immediate and follow from (4.17), also recalling (4.6), therefore we concentrate on (8.2). As for the right-hand side of (8.2), integration by parts yields and therefore, recalling the definition in (5.6), we have that is, the right-hand side of (8.2). As for the left-hand side, we similarly have G ε (t) and the left-hand side in (8.2) follows. We turn to (8.3). The left hand side inequality is nothing but the third inequality in (5.16) (that holds whenever |z| ≥ 0). For the right-hand side inequality, when t ≤ T we have, similarly to (5.22) When t > T , we notice that (5.14) and (8. , from which (8.3) follows again using the content of the last display.
Proposition 5. Let u ∈ W 1,γ (B; R N ) be a solution to (6.1), under assumptions set u in (4.18) for n ≥ 2. There exists a positive radius R * ≡ R * (data u , h(·)) ≤ 1 such that if r(B) ≤ R * and B ς ⋐ B ̺ are concentric balls contained in B, then holds with c ≡ c(data u ), where X(·) has been defined in (1.9).
Proof. We start considering a ball B r (x 0 ) ⋐ B (therefore it is r < 1), and prove that holds whenever κ ≥ 0, with c ≡ c(data u ). This is an analogue of (6.42) and to get it we modify the proof of Lemma 8, keeping the notation used there. Proceeding as for the bounds for (IV) z -(VI) z in Lemma 8, we have with c ≡ c(data u ). This estimate can be obtained by adapting those in (6.44)-(6.46), and also those for the terms in (6.17). One must take into account that now g 1,ε is independent of x (therefore the terms coming from the use of (5.26) and featuring h(·) disappear), and the fact that we can formally take σ ≡ 0 as the ratio g 2,ε /g 1,ε is bounded by a constant by (8.1) 1 . In turn, the last term in (8.7) involving the right-hand side f can be treated exactly as in (6.52). As for the remaining x-terms appearing in the first line of (8.7), with the help of (5.12) 3 , (8.1) and (8.2), we estimate for c ≡ c(data u ). Merging the content of the above three displays, choosingε > 0 small enough and reabsorbing terms we end up with (8.6), where c ≡ c(data u ). As a consequence, we proceed as for the proofs of Lemma 9 and Proposition 3. An application of Lemma 1 gives that, if B 2r0 (x 0 ) ⋐ B is any ball, then holds provided x 0 which is a Lebesgue point for |Du| (recall that here G ε is x-independent), where c ≡ c(data u ), and we have also used (5.18). Next, (3.1) gives that Using these informations in (8.8) yields where c, c * ≡ c, c * (data u ). By absolute continuity, we now determine the radius R * ≡ R * (data u , h(·)) mentioned in the statement in such a way that Using this and Young's inequality in (8.9), and yet recalling that r 0 := (τ 2 − τ 1 )/8, gives . Inequality (8.5) now follows using Lemma 2 with Z(t) := G ε (|Du|) L ∞ (Bt) and (8.3).
With Proposition 5 available, we can now complete the proof of Theorem 10. Arguing as in the proof of Theorem 9 in Section 7 (as usual with F j ≡ F εj and so on), and arrive up to (7.11), the analog of which in the present context is where c depends on data u (again recall the equivalence in (8.3)). The rest of the proof again follows with minor modifications to the proof for Theorem 9.

Proof of Theorem 12
We revisit the proof of Theorem 9 starting from the part concerning the a priori estimates of Section 6 in the case n > 2. In turn this uses as a preliminary result Proposition 1 for s * = 1, that works only assuming σ +σ < 1/n − 1/d. We notice that using the condition g 2 /g 1 ≤ c b in (4.23), by (5.24) we also get g 2,ε /g 1,ε ≤ c. This last piece of information in the proof of Lemma 8 yields the following simplified form of (6.42): For this, see also the proof of Proposition 5, while there is no need to use Proposition 1 in its proof. This last inequality still holds when n = 2; using it as in Lemma 9 yields for every κ ≥ 0, which is in fact the analog of (6.53) in the present setting. From this last inequality we again arrive at (6.58) exactly as in the proof of Proposition 3 (needless to say, for different exponents θ and β). Here the key observation is that we do not have to verify the two conditions in (6.64), that in fact are not occurring. It is only the second, and more restrictive one, that requires the stronger bound in (4.13), that can therefore be relaxed in (4.23). Starting from this fact, the rest of the proof is the same as the one of Theorem 9 in the case n > 2. In the case n = 2 the argument is similar. First, we use Lemma 10, again with s * = 1. Next, we apply this time (9.1) (instead of (6.53)) in Proposition 5 thereby getting (6.75) with G ε (·, |Du|) L ∞ (Bτ 2 ) replaced by 1. From this point on, the proof goes as in the previous case; the only remark is that now the first condition in (6.76) does not appear as a consequence of the new version of (6.75), and this new bound is exactly the one appearing in (4.23).
Step 1: Quantification of ellipticity. In the framework of Theorem 9, we define, for x ∈ Ω and t > 0, (here we take T = 1 and µ = 0) andḠ(x, t) = G(x, t) + 2 p/2 . We also set h(·) = |Da(·)|, σ := q/p−1, γ = p and σ = 0. We now observe that, replacing ≤ by < in the first bound from (1.19), we can immediately conclude with the local Lipschitz continuity of minima invoking Theorem 12. In order to get the delicate equality case in (1.19), we have to readapt some points from the proof of Theorem 12 using some additional results available when the peculiar structure in (1.10) is considered.
Let B ⋐ Ω be a ball with r(B) ≤ 1. As is [23,24], we say that the p-phase occurs when , respectively, with γ = p and g 1 , g 2 , g 3 defined in Step 1 and T = 1. Such a construction provides us with approximating integrands that are still of double phase type, i.e., with the same representation holding forH − B,ε (·), replacing a(·) by a i (B) in (10.5) above; here it is T ε = 1 + 1/ε. Next, we develop an intrinsic Sobolev-Poincaré inequality involving H ε (·); the main point is that the implied constants and exponents are independent of ε.
Proof. We prove (10.6), the proof of (10.7) being totally similar. We start considering the p-phase; this is when ( . Now, notice that the assumed bound in (1.17) implies p * ≤ q * < p, where b * := max {1, nb/(n+b)}, for b ∈ {p, q}. Therefore using again Sobolev-Poincaré and Hölder's inequalities, we estimate with c ≡ c(n, N, p, q, d, [a] 0,1−n/d;B ) andτ := q * /p < 1. Merging the content of the last two displays above easily yields for c ≡ c(n, N, p, q, d, [a] 0,1−n/d;B , Dw L p (B) ). We then pass to the (p, q)-phase, that is, when the complementary condition to (10.3) holds; here we follow the arguments from [24,Section 4]. It is then easy to see thatH ε (x, z) ∼H − B,ε (z) for (x, z) ∈ B × R N ×n . Moreover, recalling that H − B,ε can be written as in (10.4)-(10.5) with a(·) replaced by a i (B), it follows that t(H − B,ε ) ′′ (t) ∼ p,q (H − B,ε ) ′ (t), and this holds whenever t = ε, T ε , see Section 6. We are therefore in position to argue as in the proof of [24,Section 4], thereby again arriving at (10.8), for a different exponentτ ≡ τ (n, p, q) ∈ (0, 1). At this stage the proof of (10.6) follows merging the inequalities found in the two cases, and taking the smallest of the two exponentsτ found for the two phases.
Given the inequalities in (10.6)-(10.7), and the representation in (10.4)-(10.5), we can proceed for instance as in [31,Lemma 5] to get a global gradient integrability result; this also involves estimates as in (10.2) to treat the additional f -terms appearing here with respect to the case considered in . This involves a matching of local and up-to-the-boundary versions of Gehring's lemma (see [45]).
where u ∈ W 1,1 (Ω; R N ) is a minimizer of the functional in (1.10). Then there exists s * ∈ (1, s * * ), depending only on data h , but not on ε, such that holds for a constant c ≡ c(data h ), where s * * is the exponent coming from Lemma 12.
We finally remark that the constants appearing in (10.10) should a priori depend also on Du ε L p (B) , as Lemma 13 is involved in the derivation of Lemma 14 (see again [31]), and it is used with the choices w ≡ u ε and v ≡ u ε . Such a dependence on ε does not actually occur; indeed, adapting estimate (7.2) to the present case leads the bound as Du ε p L p (B) H(·, Du)+1 L 1 (B) , thereby reducing the dependence of the constants on data h in (10.1). This fact is indeed used to prove the intermediate last inequality in (10.10) with an ε-independent constant c. The last inequality in (10.10) is instead a direct consequence of (5.20). In turn, the last quantity in (10.10) is finite by Lemma 12.
Step 3: Completion of the proof of Theorem 3. We proceed almost verbatim as in the proof of Theorem 12. We start with the case n > 2; the only difference here is that we apply Proposition 1 with s * > 1 being equal to the higher integrability exponent found in Lemma 14. With no loss of generality we may assume that s * < min{2m(1+σ+σ), 2 * } = min{2mq/p, 2 * }, as required in (6.13). Notice that this is possible when σ +σ < s * /n − s * /d and therefore, in particular, when σ +σ ≤ 1/n − 1/d, which is the case considered here. Indeed, with the choice made in Step 1, this last condition translates in q/p ≤ 1 + 1/n − 1/d, which is (1.19) for n > 2. This is the essential point where the higher integrability estimates of Step 2 comes into the play, allowing for equality in (1.19). We then proceed exactly as for Theorem 12, as its assumptions are verified but for the equality case in (1.19), as noticed in Step 1. Proceeding as in the proof of Theorem 12 we arrive at (9.1), and all the foregoing considerations remain the same, but, in order to get a suitable a priori estimate, the term involving P h 1 must be estimated slightly differently from Proposition 3. More precisely, using (6.14) with the current choice of s * in (6.67), we finally come to the new uniform bound which replaces (6.58) and holds whenever B ς ⋐ B ̺ are concentric balls contained in B; the constant c depends on data h . Notice that we have applied the argument of Proposition 3 directly to u ε defined in (10.9). Using (10.10) in (10.11) yields for every s ∈ (0, 1). Again, the right-hand side stays finite by Lemma 12. This last estimate can be used as a replacement of (7.11) in an approximation argument which is at this stage completely similar to the one used for the proof of Theorem 9 and this completes the proof in the case n > 2. It remains to treat the case n = 2. For this we again turn to the arguments of Section 6.7, where we apply Lemma 10 with the choice of the number s * > 1 again determined in Lemma 14. Notice that such an application is legal as we are assuming that σ +σ = q/p − 1 ≤ 1/2 − 1/d, while again we may assume that s * < 2m(1 + σ +σ), as needed in (6.68). Then we proceed exactly as in the proof of Theorem 12 case n = 2, again noticing that only the second inequality in (6.76) is needed, and this leads to require that q/p < p, which is the second condition in (1.19) for the case n = 2.

10.3.
Proof of Theorems 4-5 and Theorem 7. We deduce Theorem 5, and therefore its special case Theorem 4, from Theorem 9. In exactly the same way, we can then deduce Theorem 7 from Theorem 11. As usual, we do it by making a suitable choice of the growth functions g 1 , g 2 , g 3 and of the parameters σ,σ, c a , c b , γ, T, µ. This requires some preparations; we split the proof in two steps. In the following we denote data k ≡ (n, N, k, ν, L, p m , p m , p M ), for every integer k ≥ 0. In the following, with abuse of notation, we shall indicate by ∂ zz e k (·) the Hessian matrix of z → e k (·, |z|), while e ′ k (x, t) keeps on denoting the (partial) derivative of the function t → e k (·, t).
Step 1: Computations. By induction we have for every x ∈ Ω and t > 0, where Then (10.12) gives We now come to the x-derivatives; we use properly defined, auxiliary vector fields D k , L k : Ω× (0, ∞) → R n for k ≥ 0, and the notation e −1 (x, t) ≡ t. Then, we have, for x ∈ Ω and t > 0 where, by induction, we have defined, for k ≥ 0 (10.15) Using (10.14)-(10.15), for k ≥ 1 we compute, again by induction (10.16) Finally, using (10.14)-(10.16) in (10.12), for every k ≥ 0 we conclude with As for the matrix ∂ zz e k (·) by a direct computation we have that hold for every choice of z, ξ ∈ R N ×n with z = 0 and x ∈ Ω.
Step 2: Determining g 1 , g 2 and g 3 . For every fixed k ≥ 0, the constants implied in the symbols and ≈, will depend on data k and we shall use the auxiliary functions h k (x) := k i=0 [|Dc i (x)| + |Dp i (x)|] + 1 for k ≥ 0. It follows that h k ∈ L d (Ω) for d > n, by assumptions. By (10.15) it is |D 0 (x, t)| h 0 (x)[| log t| + 1], so that induction gives holds for all k ≥ 1 .
In turn, this and (10.16) imply that holds for all k ≥ 1 .
By (10.12) it is also e ′ k (x, t) t p0(x)−1 ≤ t pm−1 for t ≈ 0. From this and (10.17) it follows there exit constants m k ≡ m k (data k ) ≥ 1 such that .
Now, let {c k , d k } be constants larger than 1, to be eventually chosen again in dependence on data k , and φ ∈ C([0, ∞), [0, 1]) be a non-decreasing function such that φ(t) = 0 for t ∈ [0, e/2] and φ(t) = 1 for t ∈ [e, ∞). With p m as in (1.20), and x ∈ Ω and t > 0, we define x, e) + (e 2 + 1) pm/2 ] for t ≥ e, in the notation of Theorem 9, therefore, it isḠ(x, t) ≈ e k (x, t) for t ≥ e. We are ready to check that g 1 , g 2 , g 3 satisfy the relations prescribed in Section 4.1 and required to meet the assumptions of Theorem 9. We take c a = 1, c b ≡ c b (data k ) ≥ 1 to be determined in due course of the proof, ν = p m − 1, γ = p m , T = e, µ = 0 moreover, any choice of small numbers σ,σ will do. By (10.12)-(10.13) and (10.18), it follows that (4.2) 2,3 are satisfied provided we take c k ≡ c k (data k ) large enough. Similarly, (4.2) 4 follows using (10.19). As t → g 2 (x, t)/g 1 (x, t) is increasing for all x ∈ Ω, it follows that (4.4) holds with c a = 1. In the same way (4.3) holds with the above choice of the parameters by (10.12). As for (4.14)-(4.16), notice that, given k ∈ N and ε ∈ (0, 1), there exists c ≡ c(data k , ε), such that Π k (x, t)t p0(x)+1 log t ≤ c[e k (x, t)] ε for t ≥ e. Using this, (4.14) follows, for all σ > 0, from (10.19) by taking h ≡ d k h k ∈ L d (Ω), provided we take d k ≡ d k (data k , σ) large enough. As for (4.15), notice that (g 2 [e k (x, t)] 1+2ε for every ε > 0 and t ≥ e; therefore (4.15) follows for every σ ∈ (0, 1), again taking c b ≡ c b (data k , σ) large enough. In the same way (4.16) follows, by eventually enlarging c b . This means that the assumptions of Theorem 9 are satisfied and the proof of Theorem 5 is complete.
11. Obstacle problems and Theorem 8 11.1. A general result. We start with the following constrained analog of Theorem 9, which will be used to ge the proof of Theorem 8: Theorem 13. Let u ∈ W 1,1 loc (Ω) be a constrained minimizer of the functional F 0 in (1.25), under assumptions set in (4.18) with g 3 , ∂ zz F being locally bounded on Ω × [0, ∞) and Ω × R N ×n , respectively, and γ ≥ 2. If ψ ∈ W 2,1 loc (Ω) with |D 2 ψ| ∈ X(Ω), then Du ∈ L ∞ loc (Ω; R N ×n ). Proof. In set we have initially assumed that g 3 was locally bounded in Ω × (0, ∞), while here we are assuming it is locally bounded in Ω × [0, ∞), that means that for every Ω 0 ⋐ Ω and b ≥ 0 we have that g 3 L ∞ (Ω0×[0,b]) is finite. This is is not really an additional assumption as this is automatically satisfied in all the cases considered for instance in Theorem 8. Notice also that D 2 ψ ∈ X(Ω) implies that Dψ is continuous in Ω and, in particular Dψ ∈ L ∞ (B; R n ) for any ball B ⋐ Ω; accordingly, we let T ψ ≡ T ψ (B) := Dψ L ∞ (B) + T + 1. In this respect, it is X(Ω) ⊂ L(n, 1), so that the continuity of Dψ follows from [67]. We now fix an arbitrary ball B ⋐ Ω with r(B) ≤ 1, we consider the family {F ε } constructed in Section 5.1. This time we take 0 < ε < min{1/T ψ , T }/4 and define the auxiliary problems The existence of u ε follows by standard theory, and the variational inequality holds for all w ∈ (u + W 1,γ 0 (B)) ∩ K ψ (B). Thanks to the γ-polynomial growth conditions in (5.13), we are now able to perform the linearization procedure used in [41, page 237] i.e., we can rearrange (11.2) in the following way: , for some measurable density θ ε : B → [0, 1]. Notice that the definition of f ε makes sense pointwise in light of the fact that ψ ∈ W 2,n (Ω) ∩ W 1,∞ (Ω) and of the discussion made at the beginning of Section 6 to prove (6.8). This implies that ∂ z F ε (·, Dψ) ∈ W 1,n (B) and the usual chain rule formula holds as in Section 6, again thanks to the results in [28]. We then define the constant (11.4) g ψ (B) := sup x∈B,t∈(0,T ψ ) This quantity is always bounded as ∂ zz F is in turn assumed to be locally bounded and γ ≥ 2. This is essentially the only place where such assumptions come into the play; moreover, here we also use the fact that g 3 is locally bounded as again described in the statement of the Theorem. Elementary manipulations based on the first property in (4.3) and (5.12) 3 , give Here we have also used that ε ≤ 1/T ψ to exploit the definition in (5.2). In turn, (11.5) gives for all balls B ⊆ B, where c ≡ c(n, γ, d, c a , c b , g ψ (B)). Since u ε verifies (11.3), the strict convexity of F ε (·) prescribed by (5.5) 4 implies that u ε is the unique solution of Dirichlet problem By (5.13) and (11.6), we see that problem (11.7) falls in the realm of those covered by Proposition 6 in Section 12 below, therefore u ε ∈ W 1,∞ loc (B) ∩ W 2,2 loc (B), which is exactly the information in (6.5) allowing to justify the all the subsequent calculations in Section 6 in view of an application to solutions to (11.7). Moreover, thanks to (11.5), the coercivity estimate (7.2) applied to the case of (11.7), becomes thanks to (11.5). Using Proposition 3 when n > 2, and Proposition 4 when n = 2 (with u ε ≡ u and f ε ≡ f ), we arrive at where we have used (11.5) and (11.8). Here c depends on data, h L d (B) and g ψ (B), but not on ε. Notice that, accordingly to the content of Section 6 and in particular recalling (6.24), estimate (11.9) does not hold for any ball, but it holds provided r(B) ≤ R * where, exactly as in (6.24), the threshold radius R * depends now also on h(·), Dψ(·) and g ψ (B). As for the dependence on this last quantity, there is no vicious circle here since the set function Ω 0 → g ψ (Ω 0 ) defined in (11.4) is obviously non-decreasing with respect to general open subsets Ω 0 ⋐ Ω (specifically, fix Ω 0 ⋐ Ω, determine g ψ (Ω 0 ) as in (11.4) and proceed for every ball B ⋐ Ω 0 with radius r(B) whose smallness now depends on g ψ (Ω 0 )). Estimate (11.9) can be now used to replace (7.11) in the approximation scheme of the proof of Theorem 9, with ε ≡ ε j . The rest of the proof now follows exactly the proof of Theorem 9 and leads to the conclusion, along with explicit a priori estimates obtainable by (11.9) after a suitable passage to the limit. Notice that also the argument of (7.10) can be repeated verbatim, as the obstacle constraint involved here is still convex. Indeed, notice that here we are not passing to the limit in the linearized problems (11.3), but rather directly in the obstacle problems (11.1); see for instance [29,Section 5.5] and [41] for more details on such approximation arguments. 11.2. Proof of Theorem 8 and additional results. The proof of Theorem 13 offers a route to get the obstacle version of all the other results presented in the unconstrained case; in particular, Theorem 8 follows. The key point is again to employ the linearization procedure used in [40,41] to pass from a variational inequality as in (11.2) to an equation as in (11.3), to which the estimates in the unconstrained case immediately apply. The whole procedure then works provided the additional assumptions γ ≥ 2 and g 3 , ∂ zz F ∈ L ∞ loc are in force as described in the statement of Theorem 13. With this path being settled, the reader can now easily obtain the constrained extensions of all the results presented in this paper. A few remarks are in order. First, notice that Theorems 1-2 and Theorems 4-5 in the constrained version, are a direct consequence of Theorem 13 and this can be checked exactly as in the unconstrained version. Next, again as for the case of Theorem 13, Theorems 10 and 12 admit a constrained reformulation. In turn, the former would imply a constrained version of Theorem 6. The latter would instead imply a first constrained version of Theorem 3, where the bounds in (1.19) appear in the <-version; see also Step 1 of the proof of Theorem 3. As for the full ≤-version in (1.19), it is then necessary to readapt to the obstacle case the arguments from Step 2 and 3 of the proof of Theorem 3 in Section 10.2, along the lines of the proof of Theorem 13. In this respect, the only worth mentioning difference is that the higher integrability lemmas 12 and 13 can be easily obtained in the setting of obstacle problems too, starting from the arguments indicated here. For this see also [14]. Finally, notice that in the case of functionals with (p, q)-growth, including the double phase one in (1.10), verifying the assumptions γ ≥ 2 and ∂ zz F ∈ L ∞ loc boils down to assume that p ≥ 2, as indeed done in Theorem 8. The additional (micro)assumption on g 3 in the statement of Theorem 13 is instead satisfied in every case.
Remark 9. In Theorem 13 we can trade the assumption γ ≥ 2 with µ > 0 (non-degenerate case). This eventually leads to constrained versions of Theorems 1-2 assuming that p > 1 instead of p ≥ 2, provided it is µ > 0.
As this holds whenever B ⋐ Ω, then Du is locally bounded.
Estimating the third integral in a standard way (see for instance in [44, pag. 395]) we get The involved constant c only depends on n, N, ν, Λ and γ and is otherwise independent of δ ∈ (0, 1). We now set M := sup δ Du δ L ∞ (B/2) + 1, which is a finite quantity by (12.9). We start considering the case γ ≥ 2, where we have where c ≡ c(n, N, ν 0 , Λ, γ, r(B)) and therefore, via Young's inequality, we get which is a uniform (with respect to δ) local bound for {D 2 u δ }: In the case 1 < γ < 2, we can argue exactly as after (12.11), but replacing µ by M , thereby getting again (12.12). Starting from (12.12), using the same approximation argument for the proof of Theorem 9 and in Step 2 here, we can let δ → 0 (via a subsequence) in (12.12) finally getting a local upper bound for D 2 u in L 2 . The assertion then follows via the usual covering argument.