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We establish the local Lipschitz continuity and the higher differentiability of vector-valued local minimizers of a class of energy integrals of the Calculus of Variations. The main novelty is that we deal with possibly degenerate energy densities with respect to the 𝑥-variable.

In this paper we are concerned with the regularity of solutions to a nonlinear elliptic system of $m$ equations in divergence form, satisfying $p$ growth from below and $q$ growth from above, with $p \leq q$; this case is known as $p, q$-growth conditions. Well known counterexamples, even in the simpler case $p=q$, show that solutions to systems may be singular; so, it is necessary to add suitable structure conditions on the system that force solutions to be regular. Here we obtain local boundedness of solutions under a componentwise coercivity condition. Our result is obtained by proving that each component $u^\alpha$ of the solution $u=(u^1,...,u^m)$ satisfies an improved Caccioppoli's inequality and we get the boundedness of $u^{\alpha}$ by applying De Giorgi's iteration method, provided the two exponents $p$ and $q$ are not too far apart. Let us remark that, in dimension $n=3$ and when $p=q$, our result works for $\frac{3}{2} < p < 3$, thus it complements the one of Bjorn whose technique allowed her to deal with $p \leq 2$ only. In the final section, we provide applications of our result., Comment: 15 pages

We prove a quantitative stability result for the Gauss mean value formula. We also show by an example that the estimate proved here is sharp.

In this paper we prove local Hölder continuity of vectorial local minimizers of special classes of integral functionals with rank-one and polyconvex integrands. The energy densities satisfy suitable structure assumptions and may have neither radial nor quasi-diagonal structure. The regularity of minimizers is obtained by proving that each component stays in a suitable De Giorgi class and, from this, we conclude about the Hölder continuity. In the final section, we provide some non-trivial applications of our results.

In this paper we give a general proof of Mean Value formulas for solutions to second order linear PDEs, only based on the local properties of their fundamental solution Gamma. Our proof requires a kind of pointwise vanishing integral condition for the intrinsic gradient of Gamma. Combining our Mean Value formulas with a "descent method" due to Kuptsov, we obtain formulas with improved kernels. As an application, we implement our general results to heat operators on stratified Lie groups and to Kolmogorov operators.

Let D be an open subset of $${\mathbb {R}}^n$$ with finite measure, and let $$x_0 \in D$$ . We introduce the p-Gauss gap of D w.r.t. $$x_0$$ to measure how far are the averages over D of the harmonic functions $$ u \in L^p(D)$$ from $$u(x_0)$$ . We estimate from below this gap in terms of the ball gap of D w.r.t. $$x_0$$ , i.e., the normalized Lebesgue measure of $$D \setminus B$$ , being B the biggest ball centered at $$x_0$$ contained in D. From these stability estimates of the mean value formula for harmonic functions in $$L^p$$ -spaces, we straightforwardly obtain rigidity properties of the Euclidean balls. We also prove a continuity result of the p-Gauss gap in the Sobolev space $$W^{1,p'}$$ , where $$p'$$ is the conjugate exponent of p.

The classical Faber-Krahn inequality states that, among all domains with given measure, the ball has the smallest first Dirichlet eigenvalue of the Laplacian. Another inequality related to the first eigenvalue of the Laplacian has been proved by Lieb in 1983 and it relates the first Dirichlet eigenvalues of the Laplacian of two different domains with the first Dirichlet eigenvalue of the intersection of translations of them. In this paper we prove the analogue of Faber-Krahn and Lieb inequalities for the composite membrane problem.

It is well known that an integral of the Calculus of Variations satisfying anisotropic growth conditions may have unbounded minimizers if the growth exponents are too far apart. Under sharp assumptions on the exponents we prove the local boundedness of minimizers of functionals with anisotropic p , q -growth, via the De Giorgi method. As a by-product, regularity of minimizers of some non coercive functionals is obtained by reduction to coercive ones.

Inverse problem results, related to densities with the mean value property for the harmonic functions, were recently proved by the authors. In the present paper we improve and extend them to the sub-Laplacians on stratified Lie groups.

In this paper we consider a class of non-uniformly elliptic integral functionals and we prove the local boundedness of the quasi-minimizers. Our approach is based on a suitable adaptation of the celebrated De Giorgi proof, and it relies on an appropriate Caccioppoli-type inequality.

We study the local boundedness of minimizers of a nonuniformly energy integral of the form ∫ Ω f ( x , D v ) d x under p , q -growth conditions of the type λ ( x ) | ξ | p ≤ f ( x , ξ ) ≤ μ ( x ) 1 + | ξ | q for some exponents q ≥ p > 1 and with nonnegative functions λ , μ satisfying some summability conditions. We use here the original notation introduced in 1971 by Trudinger [ 26 ], where λ ( x ) and μ ( x ) had the role of the minimum and the maximum eigenvalues of an n × n symmetric matrix a i j x and f ( x , ξ ) = ∑ i , j = 1 n a i j x ξ i ξ j was the energy integrand associated to a linear nonuniformly elliptic equation in divergence form. In this paper we consider a class of energy integrals, associated to nonlinear nonuniformly elliptic equations and systems, with integrands f ( x , ξ ) satisfying the general growth conditions above.

We establish the higher differentiability and the higher integrability for the gradient of vectorial minimizers of integral functionals with ( p , q ) -growth conditions. We assume that the non-homogeneous densities are uniformly convex and have a radial structure, with respect to the gradient variable, only at infinity. The results are obtained under a possibly discontinuous dependence on the spatial variable of the integrand.

We prove existence and regularity results for weak solutions of non-linear elliptic systems with non-variational structure satisfying ( p , q ) {(p,q)} -growth conditions. In particular, we are able to prove higher differentiability results under a dimension-free gap between p and q.

We give a regularity result for local minimizers $${u}:{\Omega \subset {\mathbb{R}}^3 \to {\mathbb{R}}^3}$$ of a special class of polyconvex functionals. Under some structure assumptions on the energy density, we prove that local minimizers u are locally bounded. For each component $${u^{\alpha}}$$ of u, we first prove a Caccioppoli’s inequality and then apply De Giorgi’s iteration method to get the boundedness of $${u^{\alpha}}$$ . Our result can be applied to the polyconvex integral $$\int_\Omega \left( \sum\limits_{\alpha = 1}^{3} |D u^\alpha|^{p} + |\operatorname{adj}_2 Du|^q + |\operatorname{det} Du|^{r}\right) {\rm d}x$$ with suitable $${p,q,r > 1}$$ .

We consider a map $u:\mathbb{R}^{n}\to \mathbb{R}^{m}$ , $n,m>1$ solution to a nonlinear systems of partial differential equations, or minimizer of a functional of the calculus of variations. It is well known that either the global or the local boundedness of $u$ cannot be obtained through truncation methods. This is due to the lack of maximum principles for general systems. Nevertheless in this paper we present a method for local boundedness of $u$ without assuming any condition on the boundary datum.

In this paper we present a new approach to prove Schauder estimates at the boundary for sub-Laplacian type operators in Carnot groups. While internal Schauder estimates have been deeply studied, up to now subriemannian estimates at the boundary are known only in the Heisenberg groups. The proof of these estimates in the Heisenberg setting, due to Jerison (J Funct Anal 43:97–142, 1981), is based on the Fourier transform technique and cannot be repeated in general Lie groups. After the result of Jerison no new contribution to the boundary problem has been provided. In this paper we introduce a new method, which allows to build a Poisson kernel starting from the fundamental solution, from which we deduce the Schauder estimates at non characteristic boundary points.

The Newtonian potential of an Euclidean ball $B$ of $\mathbb{R}^n$ centered at $x_0$ is proportional, outside $B$, to the Newtonian potential of a mass concentrated at $x_0$. Vice-versa, as proved by Aharonov, Schiffer and Zalcman, if $D$ is a bounded open set in $\mathbb{R}^n$, containing $x_0$, whose Newtonian potential is proportional, outside $D$, to the one of a mass concentrated at $x_0$, then $D$ is an Euclidean ball with center $x_0$. In this paper we generalize this last result to more general measures and domains.

We prove existence of \({u\in C^{k}(\overline{\Omega};\mathbb{R}^{n})}\) satisfying $$\left\{\begin{array}{ll} det\nabla u(x) =f(x) \, x\in \Omega\\ u(x) =x \quad\quad\quad\quad x\in\partial\Omega\end{array}\right.$$ where k ≥ 1 is an integer, \({\Omega}\) is a bounded smooth domain and \({f\in C^{k}(\overline{\Omega}) }\) satisfies $$\int\limits_{\Omega}f(x) dx={\rm meas} \Omega$$ with no sign hypothesis on f.

We prove boundedness of minimizers of energy-functionals, for in- stance of the anisotropic type (1) below, under sharp assumptions on the ex- ponents pi in terms of p � : the Sobolev conjugate exponent of p; i.e., p � = np n p , 1 p = 1 n P n i=1 1 pi . As a consequence, by mean of regularity results due to Lieber- man (21), we obtain the local Lipschitz-continuity of minimizers under sharp assumptions on the exponents of anisotropic growth.

We consider a non-linear system of m equations in divergence form and a boundary condition: {Sigma(n)(i=1) partial derivative/partial derivative x(i) (A(i)(alpha)(x, Du(x))) = 0, 1

In this survey we consider a general Hormander type operator, represented as a sum of squares of vector fields plus a drift and we outline the central role of the fundamental solution in developing Potential and Regularity Theory for solutions of related PDEs. After recalling the Gaussian behavior at infinity of the kernel, we show some mean value formula on the level set of the fundamental solution, which allow to obtain a comprehensive parallel of the classical Potential Theory. Then we show that a precise knowledge of the fundamental solution leads to global regularity results: estimates at the boundary or on the whole space. Finally in the problem of regularity of non linear differential equations we need an ad hoc modification of the parametrix method, based on the properties of the fundamental solution of an approximating problem.

Classical one-dimensional, autonomous Lagrange problems are considered. In absence of any smoothness, convexity or coercivity condition on the energy density, we prove a DuBois-Reymond type necessary condition, expressed as a differential inclusion involving the subdifferential of convex analysis. As a consequence, a non-existence result is obtained.

We study the existence of Lipschitz minimizers of integral functionals $$ \mathcal{I}(u)=\int_{\Omega} \varphi(x,\textrm{det}\,Du(x))\,dx,$$ where $\Omega$ is an open subset of $\mathbb{R}^N$ with Lipschitz boundary, $\varphi:\Omega\times (0,+\infty)\to [0,+\infty)$ is a continuous function, and $u\in W^{1,N}(\Omega, \mathbb{R}^N)$, $u(x)=x$ on $\partial \Omega$. We consider both the cases of $\varphi$ convex and nonconvex with respect to the last variable. The attainment results are obtained passing through the minimization of an auxiliary functional and the solution of a prescribed Jacobian equation.

Local Lipschitz continuity of local minimizers of vectorial integrals ∫Ωf(x,Du(x))dx is proved when f satisfies p−q growth condition and ξ↦f(x,ξ) is convex. The uniform convexity and the radial structure condition with respect to the last variable are assumed only at infinity.

We study the regularity of vector-valued local minimizers in $ W^{1,p}, p > 1 $, of the integral functional $ u \mapsto \int_\Omega [(\mu^2 + |Du|^2)^{p/2} + f(x, u,|Du|)]dx, $ where Ω is an open set in $ \mathbb{R}^N $ and f is a continuous function, convex with respect to the last variable, such that $ 0 \leq f(x,u,t)\leq C(1+t^p) $. We prove that if f = f(x, t), or f = f(x, u, t) and $ p \leq N $, then local minimizers are locally Holder continuous for any exponent less than 1. If f = f(x, u, t) and p < N then local minimizers are Hoolder continuous for every exponent less than 1 in an open set $ \Omega_0 $ such that the Hausdorff dimension of $ \Omega \backslash \Omega_0 $ is less than N−p.

We consider a class of degenerate Ornstein-Uhlenbeck operators in $\mathbb{R}^{N}$, of the kind \[\mathcal{A}\equiv\sum_{i,j=1}^{p_{0}}a_{ij}\left( x\right) \partial _{x_{i}x_{j}}^{2}+\sum_{i,j=1}^{N}b_{ij}x_{i}\partial_{x_{j}}\] where $\left( a_{ij}\right) $ is symmetric uniformly positive definite on $\mathbb{R}^{p_{0}}$ ($p_{0}\leq N$), with uniformly continuous and bounded entries, and $\left( b_{ij}\right) $ is a constant matrix such that the frozen operator $\mathcal{A}_{x_{0}}$ corresponding to $a_{ij}\left( x_{0}\right) $ is hypoelliptic. For this class of operators we prove global $L^{p}$ estimates ($1

The mathematical analysis to achieve everywhere regularity in the interior of weak solutions to nonlinear elliptic systems usually starts from their local boundedness. Having in mind De Giorgi’s counterexamples, some structure conditions must be imposed to treat systems of partial differential equations. On the contrary, in the scalar case of a general elliptic single equation a well established theory of regularity exists. In this paper we propose a unified approach to local boundedness of weak solutions to a class of quasilinear elliptic systems, with a structure condition inspired by Ladyzhenskaya–Ural’tseva’s work for linear systems, as well as valid for the general scalar case. Our growth assumptions on the nonlinear quantities involved are new and general enough to include anisotropic systems with sharp exponents and the p, q-growth case.

We consider the following classical autonomous variational problem minimize F (v) =\int_a^b f (v(x), v (x)) dx : v ∈ AC([a, b]), v(a) = α, v(b) = β , where the Lagrangian f is possibly neither continuous, nor convex, nor coercive. We prove a monotonicity property of the minimizers stating that they satisfy the maximum principle or the minimum one. By virtue of such a property, applying recent results concerning constrained variational prob- lems, we derive a relaxation theorem, the DuBois-Reymond necessary condition and some existence or non-existence criteria.

We consider a class of degenerate Ornstein–Uhlenbeck operators in $${\mathbb{R}^{N}}$$ , of the kind $$\mathcal{A}\equiv\sum_{i, j=1}^{p_{0}}a_{ij}\partial_{x_{i}x_{j}}^{2} + \sum_{i, j=1}^{N}b_{ij}x_{i}\partial_{x_{j}}$$ where (a ij ), (b ij ) are constant matrices, (a ij ) is symmetric positive definite on $${\mathbb{R} ^{p_{0}}}$$ (p 0 ≤ N), and (b ij ) is such that $${\mathcal{A}}$$ is hypoelliptic. For this class of operators we prove global L p estimates (1

We consider the following autonomous variational problem minimize { ∫ a b f ( v ( x ) , v ′ ( x ) ) d x : v ∈ W 1 , 1 ( a , b ) , v ( a ) = α , v ( b ) = β } where the Lagrangian f is assumed to be continuous, but not necessarily coercive, nor convex. We show that the existence of the minimum is linked to the solvability of certain constrained variational problems. This allows us to derive existence theorems covering a wide class of nonconvex noncoercive problems.

We show that local minimizers of functionals of the form ∫ Ω [f(Du(Χ}} + g(Χ, u(Χ))] dΧ, u e u 0 + W 1,p 0 (Ω), ∫ Ω [∫(Du(Χ))+g(Χ, u(Χ))] dΧ, ueu o +W 1,p 0 (ΩQ), are locally Lipschitz continuous provided f is a convex function with p-q growth satisfying a condition of qualified convexity at infinity and g is Lipschitz continuous in u. As a consequence of this, we obtain an existence result for a related nonconvex functional.

We consider the problem of minimizing autonomous, multiple integrals like ℘ $$ \min \left\{ \int_\Omega f\left(u ,\nabla u\right) dx : u\in u_0 + W_0^{1,p}(\Omega) \right\} $$ where $f: \ensuremath{\mathbb{R}}\times \ensuremath{\mathbb{R}} ^N\to [0 ,\infty)$ is a continuous, possibly nonconvex function of the gradient variable $\nabla u$ . Assuming that the bipolar function f ** of f is affine as a function of the gradient $\nabla u$ on each connected component of the sections of the detachment set $\mathcal{D} = \left\{f^{**} < f\right\}$ , we prove attainment for ( $\mathcal{P}$ ) under mild assumptions on f and f **. We present examples that show that the hypotheses on f and f ** considered here for attainment are essentially sharp.

Strongly elliptic differential operators with (possibly) unbounded lower-order coefficients are shown to generate $C_0$ semigroups on $L^p(\mathbb R^N)$, $1 < p < +\infty$. An explicit characterization of the domain is given.

We investigate the existence of heteroclinic solutions to a class of nonlinear differential equations ( a ( x ) Φ ( x ′ ( t ) ) ) ′ = f ( t , x ( t ) , x ′ ( t ) ) , a . e . t ∈ ℝ governed by a nonlinear differential operator Φ extending the classical p- Laplacian, with right-hand side f having the critical rate of decay -1 as |t| → +∞, that is f ( t , ⋅ , ⋅ ) ≈ 1 t . We prove general existence and non-existence results, as well as some simple criteria useful for right-hand side having the product structure f(t, x, x') = b(t, x)c(x, x'). Mathematical subject classification: Primary: 34B40; 34C37; Secondary: 34B15; 34L30.