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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.

The energy integral of the calculus of variations, which we consider in this paper, has a limit behavior when the maximum exponent $$q$$q, in the growth estimate of the integrand, reaches a threshold. In fact, if $$q$$q is larger than this threshold, counterexamples to the local boundedness and regularity of minimizers are known. In this paper, we prove the local boundedness of minimizers (and also of quasi-minimizers) under this stated limit condition. Some other general and limit growth conditions are also considered.

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.

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 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.

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 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.

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.

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 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.

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.