Lipschitz regularity for solutions of a general class of elliptic equations

We prove local Lipschitz regularity for local minimiser of \[ W^{1,1}(\Omega)\ni v\mapsto \int_\Omega F(Dv)\, dx \] where $\Omega\subseteq {\mathbb R}^N$, $N\ge 2$ and $F:{\mathbb R}^N\to {\mathbb R}$ is a quasiuniformly convex integrand in the sense of Kovalev and Maldonado, i.e. a convex $C^1$-function such that the ratio between the maximum and minimum eigenvalues of $D^2F$ is essentially bounded. This class of integrands inculdes the standard singular/degenerate functions $F(z)=|z|^p$ for any $p>1$ and arises naturally as the closure, with respect to a natural convergence, of the strongly elliptic integrands of the Calculus of Variations.

∀ w ∈ W 1,1 0 (O). The integrands F we are interested in satisfy for some H < ∞ λ max (D 2 F (z)) ≤ H λ min (D 2 F (z)) for a. e. z ∈ R N , (1.2) where here and in the following λ max (M ) and λ min (M ) denote respectively the minimum and maximum eigenvalues of the symmetric matrix M . In this introductory paragraph, the integrands obeying (1.2) will be called uniformly elliptic (even if the term is ubiquitous and thus may cause some confusion). As will be clarified later, most of the available Lipschitz regularity results for local minimisers prescribe that the singular set, where λ max (D 2 F (z)) blows up, and the degeneracy set, where λ min (D 2 F (z)) vanishes, are both bounded. As will be clarified in the following discussion, condition (1.2) alone instead allows for both sets to be simultaneously unbounded (even dense) and sizeable (in the sense of Hausdorff dimension). In order to motivate the uniform ellipticity condition (1.2), let us review some well known facts regarding local minimisers of (1.1). If F fulfils a polynomial upper bound, these are finite energy solutions of the corresponding Euler-Lagrange equation div (DF (Du)) = 0 (1.3) in Ω. Existence of minimisers is ensured by standard methods once a superlinearity condition on F at ∞ is imposed, however well known examples (see [21,33]) show that under these sôle conditions (sub-polynomial growth and superlinearity) they may fail to be locally bounded. Regularity of local minimisers is a classical topic dating back to Hilbert's XIX problem, solved through Schauder and DeGiorgi-Nash-Moser theories under the strong ellipticity assumption Since the solution of Hilbert's problem, much effort has been dedicated to weaken this assumption. A natural quantity arising in the aforementioned regularity theories is the ellipticity ratio (also called the linear dilatation of D 2 F (z)), namely e(z) = λ max (D 2 F (z)) λ min (D 2 F (z)) .
The closure of smooth, strongly elliptic integrands with respect to pointwise (or, equivalently, C 1 loc (R N )) convergence, when coupled with a uniform bound on the ellipticity ratio, turns out to be a cone consisting of • Affine functions, which are the extremals of the cone • Convex superlinear F ∈ C 1 loc (R N ) ∩ W 2,N loc (R N ) obeying (1.2) a. e. for some H < ∞ (see Corollary 2.7). The second type of integrands have been studied in the seminal papers [29,30] (to which we refer for further details) and are called H-quasiuniformly convex, 1 henceforth abbreviated by H-q. u. c.. They are related to the well developed theory of quasiconformal maps, as their gradient mapping is indeed quasiconformal. It turns out, quite conveniently we may say, that in dimension N ≥ 2 condition (1.2) implies not only superlinearity, but also sub-polynomial growth and strict convexity of the integrand F , granting existence of local minimisers (or actually, minimisers under various boundary conditions) in W If F is q. u. c., the previous sets are related to the so-called quasiconformal ∞and 0-sets respectively, appearing in the quasiconformal Jacobian problem [7]. These can be quite large, as shown in [30,Section 4 and 5]: given an arbitrary E ⊆ R N of Hausdorff dimension less than 1, there exist a q. u. c. F 1 such that E ⊆ Sing F 1 and a q. u. c. F 2 such that E ⊆ Deg F 2 . In particular, both Sing F and Deg F can be dense in R N for N ≥ 2. On the other hand, for F q. u. c., neither Sing F nor Deg F can contain rectifiable curves (see [7,30]).
Despite the natural appearance of q. u. c integrands as limits (in the sense described above) of strongly elliptic ones, the corresponding regularity theory for (1.4) seems more delicate. Indeed, the L ∞ and C α bounds for Du when F is strongly elliptic essentially depend on the quantity sup Ω λ max (D 2 F (Du)) inf Ω λ min (D 2 F (Du)) rather than on the actually controlled quantity sup Ω λ max (D 2 F (Du)) λ min (D 2 F (Du)) and a naive limiting argument is bound to fail.
Nevertheless, some classes of q. u. c. where the standard regularity theory can still produce results are already well known. An important type of q. u. c. integrands are for instance those with Uhlenbeck structure, i. e. depending only on the modulus of the gradient. Given a non-decreasing G : [0, ∞[→ [0, ∞[, the function F (z) = G(|z|) is q. u. c. if and only if G ∈ C 1 ([0, ∞)) ∩ W 2,∞ loc (R + ) and there exists a constant C > 0 such that which, since obviously Sing F ∩ Deg F = ∅ for a q. u. c. F , justifies the traditional dichotomy between the singular or degenerate case of (1.3). The regularity theory for solutions of (1.3) when F is a Uhlenbeck q. u. c. integrand is well developed, even when non-homogenous terms with low summability are included and no variational setting it available. See e. g. [12] and the literature therein for a survey on the available results for solutions of div G ′ (|Du|) |Du| Du = f (1.8) up to the Lipschitz scale, depending on the summability properties of f . Regarding higher regularity, in [32] minimisers have been proved to be C 1,α for integrands which are even more general than the Uhlenbeck ones, but still fulfil a uniform radial bound on the ellipticity ratio. By this we mean that the main assumption for the C 1,α regularity is the existence of two radial functions λ min , λ max : R + → R + obeying λ min (|z|) ≤ λ min (D 2 F (z)), λ max (|z|) ≥ λ max (D 2 F (z)) and sup t∈R + λ max (t) λ min (t) < ∞ so that these integrands are q. u. c.. In this framework both Sing F and Deg F are anyway restricted to have radial symmetry, but since neither of those can contain circles, (1.7) holds again.
Notice that when F is a q. u. c., normalised in such a way that min R N F = F (0) = 0 (which can always be safely assumed), both F and DF actually fulfil a two-sided isotropic control of the form Section 3) but, as mentioned before, no such isotropic control is available at the level of the ellipticity ratio for a general q. u. c. integrand. Our interest in this framework is motivated by the results in [4,Chapter 16] where regularity theory for the corresponding minimisers has been successfully developed in two space dimensions. More precisely, in [4,Theorem 16.4.5], the authors prove C 1,α regularity of finite energy 2 solutions of div A(Du) = 0 (1.9) in the plane, under the even more general assumption that the mapping A : for some δ > 0. Such mappings are called δ-monotone and have been introduced in [29], where it is shown that any gradient map of a quasiuniformly convex integrand is δ-monotone, but notice that in general equation (1.9) may be non-variational. Hence the results of [4] in the plane are more general and stronger than ours, but the method developed therein is based on the reduction of (1.9) to a first-order system of Beltrami type for the complex gradient of u, and thus are constrained to the two-dimensional setting.
We can now state our main result.
a. e. and fulfil the normalisation (1.10) Then any local minimiser u for J in Ω is locally Lipschitz. In particular, there exists a constant (1.11) Remark 1.2 (Comments on the statement).
• Except for the normalisation condition (1.10), the assumptions on F can be equivalently stated as F being H-quasiuniformly convex, or DF being a quasiconformal map with maximal linear dilatation bounded by H. Notice that, thanks to the quasiconformality of DF , the W 2,2 loc (R N ) regularity of F together with condition (1.2) automatically improves to F ∈ W 2,N +ε loc (R N ) for some ε > 0.
• The normalisation condition (1.10) is made only to have the cleaner estimate (1.11), as any H-q. u. c. integrand F has a unique minimum pointz. Then, the integrand is still H-q. u. c. and the functionũ( • We could also have treated local minimisers of for sufficiently smooth f . However, our method seems to provide Lipschitz regularity under non-optimal regularity conditions on f . The natural condition ensuring boundedness of Du should be f ∈ L q loc (Ω) for some q > N , but it appears that substantial modifications to the proof are needed to obtain such a result.
1.2. The rôle of the Uhlenbeck structure. Let us discuss briefly the rôle of a radial control on the ellipticity ratio in proving a-priori gradient bounds for finite energy solutions of (1.3) with a convex F . The standard strategy to this end, which goes back to Bernstein, is to find a suitable coercive function G : R N → R and coefficients a ij : R N → R such that (1) The matrix A(z) = (a ij (z)) has controlled ellipticity for every z ∈ R N ; (2) The function G(Du) solves div (A(Du) D(G(Du))) ≥ 0. (1.12) Once these tasks are achieved, linear regularity theory can be applied to G(Du). If, for instance, A turns out to be strongly elliptic, then solutions of (1.12) fulfil and we are reduced to control the integrand on the right (or variants of it) by F (Du). More generally, once suitable refinements of the linear elliptic regularity step are developed, this scheme is flexible enough to deal with choices of A which may not be strongly elliptic, as long as their ellipticity ratio does not blow up too fast for z → ∞ (see e. g. [6] and the literature therein for the non-uniformly elliptic setting). For the sake of this discussion, we will suppose that F and u are smooth and that we have chosen λ min , λ max : R N → R + in such a way that for all z ∈ R N it holds The standard approach to construct the couple (A, G) is to multiply each equation in (1.4) by ∂ α u and sum, to obtain where the last inequality follows from the convexity of F . With this choice, however, we are committing to a radial control on the ellipticity ratio, since and a natural choice for A is which is strongly elliptic if the ellipticity ratio is uniformly bounded, while in the meantime (1.14) reads div A(Du) λ min (Du) D |Du| 2 2 ≥ 0.
In order to construct G we must impose but this relation forces G, and thus λ min , to be radial functions. If this is so, i. e. λ min (z) = λ min (|z|), we can indeed set which fulfils (1.15). Notice finally that, since F (0) = 0 and DF (0) = 0, for |ω| = 1 and t > 0 it holds which can be used to bound G(Du) in terms of the natural integrand F (Du). 3 Without a radial control on λ min or λ max , however, the previous approach fails.
Remark 1.3. It may be worth noting that multiplying (1.4) by ∂ α F (Du) and summing up, one finds div D 2 F (Du)D(F (Du)) = Tr D 2 F (Du) D 2 u D 2 F (Du) D 2 u). As will be quantified in the following paragraph, the convexity of F ensures that the right-hand side above is still nonnegative, so that we have found another couple (A, G) fitting the previous scheme, namely, A = D 2 F (Du), G = F. The issue, however, is that A fails to be strongly elliptic and is only uniformly elliptic. Needless to say, the linear theory for solutions of div (A(x) Dv) = 0 under the uniform ellipticity condition a. e.
is very poor (as can be seen by simple one-dimensional examples) and does not provide boundedness of solutions. Nevertheless, estimate (1.11) says that the function v = F (Du) behaves "as if" the operator div D 2 F (Du) Dv) is strongly elliptic on v.

1.3.
Outline of the proof. The possibility of tackling the regularity problem for finite energy solutions of (1.3) in dimension N ≥ 3 under the general uniform ellipticity condition (1.2) has been considered for the first time in [23]. The approach adopted therein can be seen in the framework of nonlinear differential inclusions, originally rooted in the works [5,18,37] and well developed nowadays, thanks to the significant applications investigated in [15,28,34,39], to name a few. Specifically, one can look at (1.3) focusing only on the stress field DF (Du) = V . Formally deriving the equation, we find the system Div (DV t ) = 0 (1. 16) where Div is the row-wise divergence operator acting on matrix-valued functions and M t denotes the transpose of the matrix M . Notice that, contrary to the Laplacian operator ∆V = Div (DV ), the operator in (1.16) is far from being elliptic: its kernel contains all compactly supported solenoidal vector fields and there is no hope to prove regularity of solutions to system (1.16).
Ellipticity, however, can be restored by constraining DV to pointwise belong to a suitable cone. The natural and fruitful one in our framework, as found in [23], is given by for H given in (1.2). The fact that DV must belong to such a cone depends only on the structural condition i. e. DV must be the product of a positive definite symmetric matrix with linear dilatation bounded by H with a symmetric matrix (see Lemma 5.1 below). Note that, as H ց 1, we have K H ց K 1 = Sym N , that is the linear space of symmetric N × N matrices, hence the solutions of the differential inclusion Div (DV t ) = 0 are just the gradients of harmonic functions.
For H > 1, system (1.17) is elliptic at the L 2 level, but this kind of restored ellipticity has a limited effect at finer regularity scales. Indeed, stress fields of p-harmonic equation certainly solve the previous nonlinear differential inclusion for p = 1 + 1/H, and for p = 2 there exists (see [27]) in the plane a p-harmonic function whose stress field vanishes only at the origin and is homogenous of degree Since the last expression is always less than 1 for H > 1, the best regularity one can expect from solutions of (1.17) is at most C α , for α = α(H) ∈ [1/3, 1[ and for any H > 1. We stress here that we were not able to prove that solutions of (1.17) are bounded, which would imply the qualitative part of Theorem 1.1 and would be meaningful in light of current lines of research on differential inclusions (see e. g. [2,24]). In fact, we had to resort to additional structure possessed by the original equation (1.3). More precisely, by still denoting V = DF (Du) for a solution of (1.3) and by using the differential inclusion DV ∈ K H , we prove a family of Caccioppoli inequalities of the form for arbitrary k ∈ R and for R > r > 0. Then we translate this vectorial Caccioppoli inequality into a family of Caccioppoli inequalities on suitable scalar functions intrinsically defined as suitable Minkowski functionals associated to (which is well defined since DF is invertible), namely, for any k ∈ R. Notice that, since V = DF (Du), then and the quasiuniform convexity of F is pivotal to prove that (1.18) translates tô (1.20) Additional fine properties of the family of 1-homogeneous functions {g k } allow to adapt the classical De Giorgi method for proving boundedness of subsolutions. Notice that (1.20) exhibits two main differences with respect to a standard Caccioppoli inequality. On the one hand, due to the vectorial nature of (1.18), its dependence on the level k is encapsulated rather implicitly in the family of 1-homogeneous functions {g k } rather than directly on V . More substantially, its right-hand side is quite bigger than the one usually found for scalar problems, which would provide an integrand of the form (g k (V ) − 1) + on the right instead of g k (V ). Nevertheless, the De Giorgi method can still be adapted to this weaker setting, providing an L ∞ bound on F (Du) in terms of the L 2 norm of V . It is quite fortunate that this estimate implies, through a refinement of the results in [23], the natural bound (1.11).
1.4. Related results. In the uniformly elliptic setting the result in Theorem 1.1 has, as already remarked, been previously obtained in [32] under an Uhlenbeck type control of the form (1.13), with λ min (z) and λ max (z) depending only on the modulus of z. In [32], actually, C 1,α regularity is proved under these assumptions. Lipschitz regularity has been proved in [13] for solutions of (1.8) coupled with Dirichlet or Neumann boundary conditions, with optimal regularity assumptions both on f and on the domain. The corresponding regularity for solutions of systems with Uhlenbeck structure is treated in [14]. Notice that in [12,14,32] the authors assume that the function G ′ appearing in (1.8) (or in the lower/upper controls on D 2 F (z)) belongs to C 1 (R + ), while we admit G ′ ∈ Lip loc (R + ) (this regularity seems optimal due to [23, Example 3.5]). However, inspecting the proofs in [12,14] shows that the results therein hold under this more general assumption.
Our result covers in particular the Finslerian anisotropic setting, which we will now briefly describe due to its relevance in recent research trends. The integrand considered in this framework are of the form where G ∈ C 1 (R + ) ∩ W 2,∞ loc (R + ) is increasing, convex and fulfils (1.6) (see [23, Example 3.7]), while h is a C 2 (R N \ {0}) convex, positive, 1-homogeneous function (not necessarily symmetric) such that the principal curvatures of ∂{h < 1} are bounded from below by a positive constant, see [23, Example 3.7] for more details. The arguments in [11,Section 3] show that C 1,α regularity holds true for the corresponding minimisers when G has more stringent controls of p-growth type and h ∈ C 3,α (R N \ {0}).
While the non-uniformly elliptic case is not treated in this research, it is worth mentioning that there are many instances where Lipschitz regularity can be obtained even if the ellipticity ratio is unbounded. Generally speaking, in order to get Lipschitz continuity of local minimisers, one usually requires a growth control on the ellipticity ratio outside a bounded set, but in most cases the resulting Lipschitz bound critically depends on the diameter of the aforementioned set. The literature in this area is huge and, regarding local minimisers, we refer to [6] for some recent results and for a rather comprehensive description of this research topic. Lipschitz regularity can be obtained for rather wild functionals by the so called Hilbert-Haar method, adapted to the Calculus of Variations by Stampacchia and Hartman [25,36]. This allows to obtain Lipschitz regularity of minimisers having a prescribed boundary value obeying the so-called bounded slope condition, under very loose conditions on the integrand. We refer to [8] and the literature therein for more details on this approach. Another class of non-uniformly elliptic integrands are the so-called orthotropic ones, where both Sing F and Deg F are unbounded, being union of hyperplanes {z i = 0}. In this setting, assuming a-priori boundedness of the minimiser is the key to infer Lipschitz regularity. We refer to [9] and the literature therein for further details on this class of integrands.
1.5. Structure of the paper. In Section 2 we collect the properties of quasiuniformly convex functions which are relevant to the proof. Section 3 is devoted to a refinement of the Sobolev regularity of the stress field V = DF (Du), originally obtained in [23]. In Section 4 we construct a sequence of approximating elliptic problems and corresponding solutions, allowing to reduce the proof of Theorem 1.1 to the smooth setting. In Section 5 we prove the Caccioppoli inequality (1.18), while in Section 6 we derive several properties of the family of 1-homogeneous functions defined in (1.19). The final Section 7 is devoted to the proof of Theorem 1.1.
Acknowledgments. We thank prof. L. Kovalev for improving a preliminary version of the paper. The authors are member of GNAMPA of INdAM. G. M. acknowledges the support of DFG via grant GZ: MA 10100/1-1 project number 496629752. S. M. is partially supported by the projects PIACERI linea 2 and linea 3 of the University of Catania and by the GNAMPA's projects Equazioni alle derivate parziali di tipo ellittico o parabolico con termini singolari and Problemi ellittici e parabolici con termini di reazione singolari e convettivi.
Notations. In the whole paper we restrict to the case N ≥ 2. By |v| we denote the Euclidean norm of a vector v ∈ R N , while (v, w) stands for the scalar product of any v, w ∈ R N . Given a vector field, upper and lower indexes stand for its components and its derivatives, respectively. We sum over repeated indexes. With the symbol Ω we mean a bounded, open subset of R N , while B r (x 0 ) denotes a ball with center x 0 ∈ R N and radius r > 0, and by B r we indicate a ball of radius r, not necessarily centred at the origin. For any measurable E ⊂ R N , |E| denotes its N -dimensional Lebesgue measure. We will omit the domain of integration when it is the whole R N , if this causes no confusion. Furthermore, for the sake of notational simplicity we set Let M = (m ij ) be an N × N matrix with real entries, and let M t denote its transpose. We consider the Frobenius norm . We further denote by Id the identity matrix. Finally, for any matrix M , σ max (M ) and σ min (M ) denote its maximum and minimum singular values (i. e. the square roots of the eigenvalues of M M t ), respectively. If M is symmetric and non-negative definite, we will use the notation λ max (M ) and λ min (M ) as in this case eigenvalues and singular values coincide.

Quasiuniformly convex integrands
Let R N denote the one-point compactification of R N . By [38,Theorem 17.3] any quasiconformal map Φ : R N → R N can be extended to a homeomorphism of R N by setting Φ(∞) = ∞, in the meantime keeping its (geometric) quasiconformality constant K unaltered. Moreover, Φ ∈ W 1,N loc (R N ) and is quasisymmetric, that is, there are an increasing homeomorphism η : R + → R + and a constant C > 0 such that By [1, Theorems 3.18 and 5.1], any K-quasiconformal map Φ fulfils (2.1) with The maximal linear dilatation of Φ is defined as the (finite) number ) .
Elementary linear algebra shows that such a quasiconformal Φ is actually for all z 0 ∈ R N and all z, w ∈ R N \ {z 0 }. Notice that the constant C actually depends on N as well. Finally, recall that if Φ is quasiconformal then so is Φ −1 and the maximal linear dilatations of Φ and Φ −1 coincide. Hence (2.3) holds true for Φ −1 as well.
Kovalev's theorem [29] shows that any non-constant δ-monotone map is quasiconformal. In particular it is a homeomorphism and, directly from the definition, its inverse is δ-monotone as well. Kovalev theorem has a quantitative version proved in [3,Theorem 1]. It states that any δ-monotone map has a maximal linear dilatation obeying and this bound is sharp, i. e. it reduces to an equality for the δ-monotone linear map v → A v, where The opposite implication is in general not true. To see this, consider the linear map v → B v, with B given by which is quasiconformal with H = 3 but not even monotone. However, full (even quantitatively) equivalence of the two concepts holds true in the class of gradient mappings.
if it is convex and its gradient map is quasiconformal, with maximal linear dilatation bounded by H.
More explicitly (and by making use of Aleksandrov's theorem), F is H-q. u. c. if (i) F is convex, C 1 and W 2,1 loc (R N ), and not affine; for a. e. point z of second order differentiability.
By [30, Theorem 3.1 and Lemma 3.2], any q. u. c. function is strictly convex and coercive, thus it has a unique minimum point. Since, given two symmetric matrices M 1 , M 2 , it holds the set of H-q. u. c. functions turns out to be a convex cone, i. e. it is closed by sum and positive scalar multiples. Moreover, it is also closed by isometric change of variables, as well as dilations.
Finally, summing an affine function to a H-q. u. c. function still gives a H-q. u. c. function. For these reasons we will frequently normalize q. u. c. integrands by requiring that and simply by considering . Some examples of q. u. c. integrands have already been discussed in the previous section. Here, given two points z 1 = z 2 in R N , we construct a q. u. c. function F such that Sing F = {z 1 } and Deg F = {z 2 }, the latter sets being as in (1.5).
Without loss of generality, assume z 1 = 0 and z 2 = w, for fixed w ∈ R N , and set r = |w|.
so that a direct computation shows (cf. [23, Example 3.6]) and consider the (positive and finite) numbers which shows through (2.10) that and λ min (D 2 F (z)) → 0 for z → 0. From (2.6) we infer that for all z ∈ B r/4 (w) Many properties of q. u. c. functions have been studied in [29,30], to which we refer for further details and characterisations. Here we gather the ones that are needed in the proof of our main result. We start by a converse of Kovalev's theorem relating the δ-monotonicity to the quasiconformality in a quantitative form.
and the bounds are sharp.
Proof. The fact that if DF is δ-monotone and non-constant then F is H-q. u. c. has been proved, as mentioned before, in [3, Theorem 1]. To prove the opposite implication, suppose F is H-q. u. c., so that DF is quasiconformal. By [29, Section 3], the δ-monotonicity of DF is equivalent to at a. e. points of second order differentiability where (2.5) holds true. Note that we can assume that D 2 F is symmetric, as this follows from Alexandrov's theorem (see [35,Corollary 2.9]), and that it is strictly positive definite, thanks to the quasiconformality of DF . Since (2.12) is invariant by orthogonal change of variables, we can assume that D 2 F is diagonal with positive eigenvalues λ 1 < λ 2 < · · · < λ N . We are therefore reduced to find, for v = (v 1 , . . . , v N ), the value of In order to determine the latter we invoke Cassels inequality (see [40,Appendix]), which reads as for any choice of w i ≥ 0 not all identically equal to zero and a i , b i > 0 such that Indeed, it suffices to choose b i = 1, a i = λ i and w i = v 2 i to obtain that and actually the equality holds true for the vector Since λ N ≤ H λ 1 by assumption and the map t → 2 √ t/(t + 1) is decreasing, we infer that so that (2.12) is proved for δ = 2 √ H/(H + 1), as claimed. The optimality of this estimate follows by choosing F (z) = (A z, z) with A given in (2.4) (here we use the fact that A is symmetric).
For further reference, we regroup the previous discussion and some other useful properties of H-q. u. c. functions in the following proposition.
Proposition 2.6. Let F : R N → R be a H-q. u. c. function. The following holds: (1) F is strictly convex, coercive and W 2,N loc (R N ).
Proof. The first two properties have already been discussed, so we focus on (3). We will prove it by exploiting the δ-monotonicity and the quasisymmetry of DF . To this aim we note that, by considering F (z +z) − F (z), we can assumez = 0 and F (z) ≥ F (0) = 0. To prove (2.13), we start by noticing that the δ-monotonicity of DF gives On the other hand, inequality (2.3) applied to DF with z 0 = 0 (and since DF (0) = 0, too) gives so that (2.16) can be estimated as Inequality (2.14) is proven similarly. It holds Thanks to the quasisymmetry (2.3) of DF we get whenever |x| ≤ |w|, and thus (2.14) follows. The first part of assertion (4) has been proved in [23,Proposition 2.3], while the C 1 loc (R N ) convergence of F n to F is trivial. Since DF n are homeomorphisms which are locally uniformly converging to DF , the same is true for DF −1 thanks to Arens' theorem (see [17]). Finally, part (5) has been proved in [23, Proposition 2.3-(iv)], where it in particular it is shown that if P δ = Id + δ DF −1 −1 , for a. e. z ∈ R N it holds .
the claimed bound follows.
is closed with respect to point-wise a. e. convergence and the smooth, strongly elliptic integrands in C H are dense in it. Moreover, point-wise a. e. convergence of (F n ) n in C H implies C 1 loc (R N ) convergence of (F n ) n and L 1 loc (R N ) convergence of (det D 2 F n ) n . Proof. By [30, Lemma 2.5] we know that the point-wise limit of q. u. c. functions is either q. u. c. or affine and that the point-wise a. e. convergence implies the C 1 loc (R N ) one. Therefore it suffices to show that is lower semicontinuous with respect to C 1 loc (R N ) convergence on the cone of q. u. c. functions. Let F n ∈ C H be such that F n → F in C 1 loc (R N ) with none of the F n affine. By Lemma 2.5, DF n is 2 √ H/(H + 1)-monotone hence, by passing to the limit in the definition od δmonotonicity, so is DF . By Lemma 2.5 again, this implies that F is H-q. u. c., giving the claimed lower semicontinuity.
To prove the density statement, let F be H-q. u. c. for some H. For δ n , ε n , µ n ↓ 0, the functions F n (z) = F δn * ϕ εn (z) + µ n 2 |z| 2 (here F δn denotes the Moreau-Yoshida regularisation of F ) constructed through Proposition 2.6, points (4) and (5), are H-q. u. c. integrands that approximate F point-wise. Using (2.6) together with basic properties of convolution and Proposition 2.6-(5), yields for all z ∈ R N , hence F n are strongly elliptic integrands. If instead F is an affine function of the form F (z) = (w, z) + c for fixed w ∈ R N and c ∈ R, we can consider G(z) = |z| 2 /2 and set F n (z) = n G(w + z/n) − G(w) + c, DF n (z) = DG(w + z/n), D 2 F n = Id n .
Clearly F n → ℓ in C 1 loc (R N ) and the F n are 1-q. u. c. and strongly elliptic. The last stated property on the L 1 -convergence of the determinants is contained in [30, Lemma 2.5].
Actually (see again [30, Lemma 2.5]), the pointwise convergence in the previous corollary can be weakened to hold on a dense subset of R N .
Remark 2.8. The previous corollary, as outlined in its proof, implies the lower semicontinuity of the maximal linear dilatation of monotone, quasiconformal gradient maps. Lower semicontinuity of the maximal linear dilatation in the larger class of quasiconformal maps holds true for N = 2, but fails for N ≥ 3, due to a famous counterexample of Iwaniec [26].
We conclude this section by introducing a generalisation of the function η H given in (2.2), which will appear in many subsequent computations. Given a, b > 0, the auxiliary functions For all s, t ≥ 0 (3) For all t > 0 and σ > 0 it holds

20)
for positive constants C = C(a, b, σ). The same estimate holds true for η −1 a,b as well.
The proof of these facts is elementary and is therefore omitted.

Local minimisers and Sobolev regularity of their stress field
If F is C 1 , strictly convex and coercive, let {z} = Argmin (F ). It is readily checked that any local minimiser u of J defines a local minimiserũ(x) = u(x) − (z, x) for the functionalJ given as in (3.1), but withF (z) defined as in (2.9). In particular, we can assume that (2.7) holds true, i. e. thatz = 0. If F is q. u. c., which will be assumed henceforth, using (2.14) for z = 0 gives so that for all |z| > 1 it holds In particular, any local minimiser for J in Ω belongs to W (Ω). More precisely, despite the fact that q. u. c. integrands are per se anisotropic, they enjoy, together with their derivative, an isotropic growth control allowing to treat minimisation problems for J in standard Orlicz-Sobolev spaces. Indeed, after normalising F so that (2.7) holds true, we can let for t ≥ 0 Notice that a is continuous on [0, ∞[ and positive for t > 0. Given is an open neighbourhood of the closed set DF (B t 1 ), thus a(t 2 ) > a(t 1 ), i. e. a is strictly increasing. By the quasisymmetry of DF , choosing z 0 = 0 and |w| = t in (2.3), we obtain that for any z ∈ R N it holds so that for any k ≥ 0, t ≥ 0, the monotonicity of η H grants a(k t) ≤ C η H (k) a(t). while the δ-monotonicity and quasisymmetry of DF give The map DF enjoys similar isotropic bounds, namely, where the first inequality follows from (2.3) as before, and it is furthermore possible to show that 1 C A(|z|) ≤ A * (DF (z)) ≤ C A(|z|) Using the fact that O has finite measure and A satisfies the ∆ 2 condition, the summability of A(|Du|) can be equivalently stated as Thanks to the ∆ 2 condition on A, the so-called Orlicz-Sobolev space W 1,A (O) so defined turns out to be a Banach space. Existence of local minimiser given, say, a boundary datum ϕ ∈ W 1,A (Ω) for Ω bounded is thus granted by minimising J on u + W 1,A 0 (Ω), the latter space being the closed vector space of those v ∈ W 1,A (Ω) whose extension at zero outside Ω belongs to W 1,1 (R N ). Indeed, thanks to the validity of the ∆ 2 condition on both A and A * , the Banach space W 1,A 0 (Ω) is reflexive by [ Notice that for any u ∈ W 1,A (O) the inequalities in (3.5) imply that |DF (Du)| ∈ L A * (O) (thus a fortiori |DF (Du)| ∈ L 1 (O)) and therefore Hölder's inequality in Orlicz spaces ensures that We summarise the previous discussion in the following proposition. The next proposition is essentially contained in [23, Theorem 3.3], but we will need the more explicit estimates given in its present form. It revolves around the Sobolev regularity of the stress field V (x) = DF (Du(x)). (3.7) Proposition 3.3. Let F be a H-q. u. c. function obeying (2.7) and let i F be as in (2.8). Let u be a local minimiser for J, and let V be given by (3.7). Then, V ∈ W 1,2 loc (Ω) with estimates for a positive constant C = C(N, H) and all balls B R such that B 2R ⊆ Ω.
Proof. From [23, Theorem 3.3, (3.10)] we know that for a constant C = C (N, H, R). More precisely, setting for a constant C which henceforth depends only on H and N . On the other hand, by Poincaré inequality and this last estimate we have We claim that for a constant C = C (N, H). To this aim, for any t > 0 we proceed as followŝ (3.13) The first term on the above right-hand side is estimated by using the quasisymmetry of DF while on the second term we use (2.13). Therefore (3.13) reduces tô Choose t such that Finally, notice that t η 1 for all t ≥ 0, so that taking (3.9) and (3.7) into account implies the validity of (3.12). Recalling (3.10) and (3.11) finally gives (3.8a) and (3.8b), respectively.

A regularising procedure
As customary, it will be convenient to work under the assumption that both the integrand and the minimiser under scrutiny are smooth. In order to do so, we introduce a family of regularised problems having the required smoothness and whose integrands and corresponding minimisers converge to the original problem.  Regularised problems). Let F be a H-q. u. c. function fulfilling (2.7) and let u be a local minimiser for J in W 1,1 (Ω). Then, for any ball B 2R ⋐ Ω there exists a sequence F n ∈ C ∞ (R N ) of H-q. u. c. integrands obeying (2.7) and inf z∈R N λ min (D 2 F n (z)) > 0 and a sequence of corresponding minimisers u n ∈ C ∞ (B 2R ) of Proof. We set for brevity B = B 2R and p = 1 + 1/H. Fix an even ϕ ∈ C ∞ c (B 1 ; [0, +∞[) such that ϕ 1 = 1 and use the notation of Proposition 2.6-(4). Notice that for any such ϕ it holds z =ˆ(z − y) ϕ(y) dy so that Jensen inequality gives with ε n ↓ 0 and µ n ↓ 0 to be chosen, so that F n fulfils for each n all the conditions stated in the lemma. Notice that thanks to (3.2) and the convergence F n → F in C 1 loc (R N ) we have respectively, for constants C independent of n. Let, for sufficiently large n, ψ n = u * ϕ 1/n ∈ C ∞ (B) and consider the minimisation problem inf{J n (w) : w ∈ Lip(B), w = ψ n on ∂B}. (4.4) According to [36,Theorem 9.2], there is a solution u n of (4.4), which (see [10, p. 5923] and (3.2)) also solves J n (u n ) = inf{J n (w) : w ∈ ψ n + W 1,p 0 (B)}. By [8,Theorem 4.1] there are constants A n ≥ 1 (depending only on B and on the regularity of ψ n , but not on ε n , µ n ) such that The integrand F is therefore strongly elliptic on the range of Du n since F n ∈ C ∞ (R N ) and λ min (D 2 F n (Du n )) ≥ µ n , λ max (D 2 F n (Du n )) ≤ C < ∞ on B, so that standard regularity theory gives u n ∈ C ∞ (B). We first choose µ n ↓ 0 so that lim n µ nˆB |Dψ n | 2 dx = 0. and observe that, since ϕ εn * F → F in C 1 loc (R N ) as ε n ↓ 0 and M n is independent of ε n , we can pick (ε n ) ⊆ (0, 1), ε n ↓ 0, so that Clearly, it still holds F n → F in C 1 loc (R N ). We claim that (4.1) holds true with such choices. By the minimality of u n , it holds J n (u n ) ≤ J n (ψ n ), while from (4.5) we get (4.7) In order to estimate the rightmost term of (4.7) we use (4.6) to havê while the vector-valued Jensen inequality implieŝ Therefore from (4.8) we have which, inserted into (4.7), gives lim n J n (u n ) ≤ J(u). (4.9) In particular, J n (u n ) is bounded and (4.3) then implies a uniform bound on Du n in L p (B). Moreover, since u n − ψ n ∈ W 1,p 0 (B), then Poincaré's inequality gives u n L p (B) ≤ u n − ψ n L p (B) + ψ n L p (B) so that (u n ) is bounded in L p (B) as well. Therefore (u n ) possesses a (not relabeled) subsequence weakly converging in W 1,p (B) to some v and it is readily checked that v ∈ u+W 1,p 0 (B). This in turn implies that (4.1b) is satisfied. Now, since the map is weakly lower semicontinuous in W 1,p (B) and F n ≥ F thanks to (4.2), we get Coupling the latter with (4.9) gives J(v) ≤ J(u), implying v = u by the strict convexity of F . In particular, up to subsequences, Therefore, from (4.10) and (4.9) we infer B F (Du n ) dx →ˆB F (Du) dx, (4.12) that is (4.1a). We now apply Proposition 3.3 to DF n (Du n ), thus obtaining where the uniform bound holds thanks to equations (4.3) and (4.12), being B = B 2R . Setting we can pick a subsequence such that V n → V weakly in W 1,2 (B R ), strongly in L 2 (B R ), and pointwise a. e. in B R , for a suitable V ∈ W 1,2 (B R ). Since DF −1 n → DF −1 locally uniformly thanks to Proposition 2.6-(4), we infer that DF −1 n (V n ) → DF −1 (V ) a. e.. Then (4.13) implies that Du n → DF −1 (V ) a. e., and in turn (4.11) allows the identification V = DF (Du). This shows (4.1c) and the proof is thus complete.

Caccioppoli inequality
In this section we prove a Caccioppoli-type inequality. To this aim, let us first show the following result, which is a simplification of [23, Lemma 3.1].
Lemma 5.1. Let P and S be N × N symmetric matrices, with P positive definite. Then, Proof. Let us observe that both sides of (5.1) are invariant by orthogonal change of basis, so we can assume that P is diagonal, with positive eigenvalues λ 1 , . . . , λ N obeying for any i, j = 1, . . . , N . Then, if S = (a ij ), we have as claimed.
Lemma 5.2 (Caccioppoli Inequality). Let F be a smooth H-q. u. c. function and let u ∈ C ∞ (B R ). For V given by (3.7), assume that Moreover, for a given affine function ℓ : R N → R, and all ρ < R, set Then, for all ρ < R, it holdŝ Proof. Let ε > 0 and set V ε = V + ε Du = DF (Du) + ε Du. Differentiating the latter gives where the matrices D 2 F (Du) + ε Id and D 2 u are symmetric positive definite and symmetric, respectively. By (2.5), we have . We want to integrate the above inequality over A(ℓ, r), for r < R, assuming for the moment that {F (Du) = ℓ(Du)} is a C 1 hypersurface in B R . Using the divergence theorem giveŝ where n denotes the exterior normal to ∂A(ℓ, r) and where we used the fact that (5.2) implies div V j = 0 for all j = 1, . . . , N . To compute the last integral in (5.5), notice that being D 2 F non-negative definite. Then the previous inequality implies that the rightmost integral in (5.5) evaluated on B r ∩ {F (Du) = ℓ(Du)} has a negative sign and hence (5.5) reduces toˆA (ℓ,r) where in the last step we used the Schwarz inequality. Let now ζ be the function given by Let ϕ ∈ C ∞ ([0, R]) with ϕ(R) = 0. We multiply (5.6) by ϕ 2 and integrate in [0, R]. Taking (5.7) into account, integrating by parts and using the fact that ϕ(R) = ζ(0) = 0, we get R 0ˆA(ℓ,r) Fix 0 < ρ < R.
A minimiser for this problem is found through elementary considerations as so that an explicit computation gives

and (5.4) follows.
To remove the assumption that the set {F (Du) = ℓ(Du)} is C 1 in B R , we invoke Sard's theorem, which ensures that this property is satisfied by {F (Du) = ℓ(Du) + k}, for a. e. k ∈ R. Given k n ↓ 0, we can repeat the same argument as before, starting from (5.5), thus arriving toˆA (ℓ+kn,r) Notice that A(ℓ + k n , ·) ց A(ℓ, ·), so we can pass to the limit in (5.8) by monotone (or dominated) convergence, and in turn (5.4) holds true also in this case.
We report the following consequence, already present in [23, proof of Theorem 3.2], for the sake of completeness. Corollary 5.3. Let F be a H-q. u. c. function, let u be a local minimiser for J in Ω in the sense of Definition 3.1 and let V be given by (3.7). Then, for C = C(H, N ) and all balls B R such that B 4R ⊆ Ω.
Proof. By using Lemma 4.1 on B 2R we can suppose that both F and u are smooth, since for the regularised problems the left-hand side of (5.9) is lower semicontinuous with respect to the weak convergence in W 1,2 (B R ), while the right-hand side is (up to subsequences) convergent. We thus apply Lemma 5.2 for ℓ(z) ≡ min F to get for all 0 < ρ < r < 2R. Let E be a continuous extension operator E : W 1,2 (B 1 ; R N ) → W 1,2 0 (B 2 ; R N ). The Gagliardo-Nirenberg inequality [31,Theorem 12.83] ensures that for C = C(N ), so that by the continuity of Ê for a bigger constant depending on the operator norms of E : For any ε > 0 we apply to the previous estimate Young's inequality with exponents (N + 2)/2 and (N + 2)/N to get which rescales tô C r 2 and apply (5.11) with U = V and such ε to (5.10) to get for all 0 < ρ < r < 2R. An application of [20,Lemma 3.1,Ch. 5] shows that then which, for r = 2R, ρ = R, provides (5.9).

A family of 1-homogeneous functions
Let F be a H-q. u. c. function. In this section we gather the basic properties of the function and of the family of Minkowski functionals related to it, namely, for any k ≥ 0. Note that, since DF is a homeomorphism of R N , the function in (6.1) is well defined.
Proposition 6.1. Let F be a H-q. u. c. function fulfilling (2.7). Then, G is coercive and for any k ≥ 0 the set {G ≤ k} is star-shaped with respect to the origin.
Proof. We apply (2.13) with the vector DF −1 (z) and use (2.7) to have Furthermore, the quasisymmetry (2.3) of DF −1 together with DF −1 (0) = 0 give Gathering the previous inequalities and making use of (2.19) and (2.17) we have thus proving the coercivity of G. In particular for any k > 0 the set {G ≤ k} is compact with nonempty interior.
Let us now show that {G ≤ k} is star shaped with respect to the origin. Since G(0) = 0 by the normalisation (2.7), it suffices to show that for any z ∈ S N −1 the function t → G(t z) is nondecreasing. To this end, suppose first that F ∈ C 2 (R N ) fulfils for some µ > 0 independent of z. Then DF is a diffeomorphism and it holds proving the claim. In the general case, thanks to Proposition 2.6-(4), we know that any Hq. u. c. function F can be approximated in C 1 by a sequence (F n ) of C 2 and H-q. u. c. functions obeying (6.3), and such that DF −1 n → DF −1 locally uniformly. Hence, G n = F n • DF −1 n → G locally uniformly, too. It follows that t → G(t z) is nondecreasing, being the pointwise limit of a sequence of nondecreasing functions. The proof is thus complete.
Remark 6.2. In general, the sets {w ∈ R N : G(w) ≤ k} = {DF (z) : F (z) ≤ k} are not convex, so that the functions g k defined in (6.2) may fail to be norms. Moreover, by employing the approximation in (2.15) in the previous proof, we see that the H-q. u. convexity assumption is not needed to prove the second statement of the previous proposition and, given any convex F ∈ C 1 (R N ) obeying (2.7), the sets {DF (z) : F (z) ≤ k}, while not convex in general, are always star-shaped with respect to the origin. Proposition 6.3. Let F be a C 2 and H-q. u. c. function obeying (2.7) and inf z∈R N λ min (D 2 F (z)) > 0.
Then, the following holds: (1) g k is a Lipschitz 1-homogeneous function with . (6.7) Proof. Since DG given in (6.4) doesn't vanish outside the origin, the implicit function theorem implies that the boundary ∂{G ≤ k} = {G = k} is a C 1 -hypersurface and therefore g k is C 1 outside the origin. By construction g k is positively 1-homogeneous, hence it fulfils Dg k (z), z = g k (z), ∀ z = 0, (6.8) Dg k (λ z) = Dg k (z), ∀ λ > 0, z = 0, (6.9) while by construction it holds From the last property, in particular, it holds g k ≡ 1 on {G = k}, and therefore the Lagrange multiplier rule gives Dg k (z) = α(z) DG(z) for some α(z) ≥ 0 and all z ∈ {G = k}. This, together with (6.8), implies that on {G = k} we have
Therefore we can use (6.13) with the vector z g k (z) , thus obtaining This proves (6.5 By the definition of G, it holds DF −1 (x), DF −1 (ȳ) ∈ {F = k}, hence (2.14) gives Inserting this estimate into (6.14) gives us the conclusion. Finally, to prove the last assertion, let us fix k > h, choosex such that G(x) = k as well as g h (x) = inf By the definition ofx andỹ and the 1-homogeneity of g h it follows that To estimate the rightmost integral in the above inequality we use (2.3) for the map DF (recalling that DF (0) = 0) and (2.18) to havê Therefore (6.18) simplifies to On the other hand, (2.13) implies h = F (y) ≥ |y| |DF (y)|/C.
Gathering the previous two inequalities gives For any k ≥ 0 we consider the Minkowski functional of {G ≤ k} defined as in (6.2), with G given by (6.1), which satisfies by construction Let V be given by (3.7). It holds ϕ (g k (V ) − 1) + ∈ W 1,2 0 (BR) and thanks to (7.4) where we set i. e. equation (5.3) for the affine function ℓ(z) ≡ k. Then the Sobolev embedding and the chain rule give In order to estimate the above right-hand side, for the first term we use (7.4) and (7.3) to achieveˆBR while for the second term we apply the chain rule and inequalities (7.4) and (6.5) to havê |DV | 2 2 dx.

(7.7)
To further estimate the last integral, we use the Caccioppoli inequality (5.4) with ρ =R < R, which results in Moreover, we observe that g k (V ) ≥ 1 on A(k, R), hence the 1-homogeneity of g k implies that (7.9) From (7.8), (7.9) and (6.6) we have that (7.7) simplifies tô which, inserted into (7.5) together with (7.6), gives Let now 0 < h < k and observe that g h (V (x)) − 1 ≥ inf {G≥k} (g h − 1) for x ∈ A(k, R), therefore (6.7) gives Inserting this estimate into (7.10) gives for 0 < h < k < 2 h ˆA (k,r) where in the second inequality we used the fact that A(k, R) ⊆ A(h, R) as well as g h ≥ g k for all h ≤ k (7.13) and the bound k h − 1 −2H ≥ 1 for 0 < h < k < 2h. Estimate (7.12) in conjunction with (7.11) allows for a De Giorgi-type iteration, whose proof we include here for completeness. For 0 < h < k < 2h, we apply Hölder's inequality, (7.12), (7.11) and (7.13) to havê A(k,r) (7.14) Let M > 0 to be chosen and set for n ≥ 0 so that lim n→∞ k n = M and lim n→∞ r n = R/2. An elementary computation shows that k n ≤ k n+1 ≤ 2 k n , k n+1 k n − 1 ≥ 1 2 n+2 , (r n − r n+1 ) 2 = R 2 4 n+2 , therefore (7.14) with the viable choice k = k n+1 and h = k n reads as Then, thanks to (2.21) and again (2.20) it is sufficient to require F (Du) dx . (7.17) Choosing M such that the equality in (7.17) holds true gives, thanks to (7.16), that F (Du) ≤ M in B R/2 , with the estimate We can then remove assumption (7.2) to get, for a general i F > 0, F (Du) dx . (7.18) We next take advantage of the invariances of this estimate. For any λ > 0, the function u λ (x) = λ u(x) is a local minimiser in Ω for the functional J λ having integrand F λ (z) = F (z/λ).
In particular, (7.18) holds true also for such F λ , u λ , that is, F λ (Du λ ) dx . (7.19) Notice that F λ is still H-q. u. c. and that For any z, w of unit norm, the distortion estimate (2.3) applied first to the vectors z/λ, w and then to w, z/λ gives DF z λ ≤ C |DF (w)| η H (1/λ) as well as |DF (w)| ≤ C DF z λ η H (λ), respectively, so that i F C η H (λ) ≤ DF z λ ≤ C i F η H (1/λ) for all |z| = 1. Taking the infimum for such z's and using (7.20) give Since DF is H N −1 -quasiconformal, it is also locally 1/H-Hölder continuous and so is the map t → |DF (t z)|. It follows that λ → i F λ is continuous on ]0, +∞[, being an infimum of equicontinuous functions. Moreover, it fulfils Thanks to the intermediate value theorem we can then choose λ so that For such λ and corresponding u λ , F λ , (7.19) becomes (7.1), since This gives us the conclusion in the smooth case.
To remove the smoothness assumption on F and u, let F n and u n be the regularised approximations obtained thanks to Lemma 4.1. Then we can apply the same argument as before to F n , u n , thus obtaining F n (Du n ) dx. (7.21) Now observe that (4.1) allows to assume (up to subsequences) that DF n (Du n ) → DF (Du) pointwise a. e. in B 2R , while Proposition 2.6-(4) ensures that DF −1 n → DF −1 locally uniformly, so that both Du n → Du and F (Du n ) → F (Du), pointwise a. e. in B R . This observation and the first estimate in (4.1) allow to pass to the limit in (7.21), thus obtaining again the conclusion.