A necessary condition for extremality of solutions to autonomous obstacle problems with general growth

TL;DR

This paper establishes a necessary condition for the extremality of solutions to autonomous obstacle problems with general growth, using convex analysis and calculus of variations.

Contribution

It provides a new necessary condition for extremality in obstacle problems with general growth conditions, expanding the theoretical understanding of such variational problems.

Findings

Characterization of solutions via primal-dual formulation

Existence and uniqueness of solutions under convexity and superlinearity

Necessary condition for extremality of solutions

Abstract

Let us consider the autonomous obstacle problem \begin{equation*} \min_v \int_\Omega F(Dv(x)) \, dx \end{equation*} on a specific class of admissible functions, where we suppose the Lagrangian satisfies proper hypotheses of convexity and superlinearity at infinity. Our aim is to characterize the solution, which exists and it is unique, thanks to a primal-dual formulation of the problem. The proof is based on classical arguments of Convex Analysis and on Calculus of Variations' techniques.