Giulia Treu

Gradient maximum principle for minima

AbstractWe state a maximum principle for the gradient of the minima of integral functionals

Existence and Lipschitz regularity for minima

I(u) = \int_\Omega{f(\nabla u)}+ g(u)]dx,{\text{on }}\bar u + W_0^{1,1} (\Omega ),

A Comparison Principle and the Lipschitz Continuity for Minimizers

just assuming that I is strictly convex. We do not require that f, g be smooth, nor that they satisfy growth conditions. As an application, we prove a Lipschitz regularity result for constrained minima.

A comparison principle for minimizers

We consider a functional I(u) = ∫Ωf(∇ u(x)) dx on u0 + W1,1(Ω). Under the assumption that f is just convex, we prove a new Comparison Principle, we improve and give a short proof of Cellinas Comparison result for a new class of minimizers. We then extend a local Lipschitz regularity result obtained recently by Clarke for a wider class of functions f and boundary data u0 satisfying a new one-sided Bounded Slope Condition. A relaxation result follows.

Absolutely continuous representatives on curves for Sobolev functions

We prove the existence, uniqueness and Lipschitz regularity of the minima of the integral functional I(u) = ∫ Ω L(x,u,⊇u)dx on ū + W 0 1,q (Ω) (1 ≤ q ≤ +∞) for a class of integrands L(x,z,p) = f(p) + g(x, z) that are convex in (z,p) and for boundary data satisfying some barrier conditions. We do not impose regularity or growth assumptions on L.

On the equivalence of two variational problems

Abstract Let be bounded, open and convex. Let be convex, coercive of order p > 1 and such that the diameters of the projections of the faces of the epigraph of F are uniformly bounded. Then every minimizer of is Hölder continuous in of order whenever φ is Lipschitz on ∂Ω. A similar result for non convex Lagrangians that admit a minimizer follows.

Lipschitz regularity for minima without strict convexity of the Lagrangian

We state some recent results on the existence, uniqueness and Lipschitz regularity of the minima of the integral functional \(I(u) = \int_\Omega L (x,u,\triangledown u)dx\) for a class of integrands L(x, z, p) = f (p) + g(x,z) that are convex in (z, p) and for boundary data satisfying some barrier conditions. We do not impose regularity or growth assumption on L.

On the Lavrentiev phenomenon for multiple integral scalar variational problems

We give some conditions that ensure the validity of a Comparison principle for the minimizers of integral functionals, without assuming the validity of the Euler-Lagrange equation. We deduce a weak maximum principlefor (possibly) degenerate elliptic equations and, together with a generalization of the bounded slope condition, the Lipschitz continuity of minimizers. To prove the main theorem we give a result on the existence of a representative of a given Sobolev function that is absolutely continuous along the trajectories of a suitable autonomous system.