Carlo Mariconda

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 ),

On a parametric problem of the calculus of variations without convexity assumptions

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 and the Lipschitz Continuity 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.

A comparison principle for minimizers

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.

An Elementary Proof of a Characterization of Constant Functions

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.

Absolutely continuous representatives on curves for Sobolev functions

Abstract The parametric integral I ( C ) = ∝ a b f ( x ′( t )) dt attains the minimum in a class of rectifiable curves C : x = x ( t ), a ⩽ t ⩽ b , under slow growth conditions and no convexity assumption on f.

Generating Formal Series and Applications

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.