Maximum Principles for Vectorial Approximate Minimizers of Nonconvex Functionals

TL;DR

This paper develops maximum principles for vectorial approximate minimizers of nonconvex functionals, notably removing the quasiconvexity assumption by using a novel nonlinear convergence method involving reflections and invariance properties.

Contribution

It introduces a new approach to establish maximum principles for nonconvex functionals without quasiconvexity, expanding the applicability of such principles in calculus of variations.

Findings

Maximum principles apply to vectorial approximate minimizers.

The quasiconvexity assumption is no longer necessary.

The method handles non-existence of minimizers and solutions of PDEs.

Abstract

We establish Maximum Principles which apply to vectorial approximate minimizers of the general integral functional of Calculus of Variations. Our main result is a version of the Convex Hull Property. The primary advance compared to results already existing in the literature is that we have dropped the quasiconvexity assumption of the integrand in the gradient term. The lack of weak Lower Semicontinuity is compensated by introducing a nonlinear convergence technique, based on the approximation of the projection onto a convex set by reflections and on the invariance of the integrand in the gradient term under the Orthogonal Group. Maximum Principles are implied for the relaxed solution in the case of non-existence of minimizers and for minimizing solutions of the Euler-Lagrange system of PDE.