Existence of an absolute minimizer via Perron's method

TL;DR

This paper proves the existence of an absolute minimizer for a supremal functional using Perron's method, under quasiconvexity and coercivity assumptions, extending previous results in the calculus of variations.

Contribution

It introduces a new proof of the existence of absolute minimizers for supremal functionals via Perron's method, completing prior work by other researchers.

Findings

Established the existence of absolute minimizers under quasiconvexity and coercivity.

Extended previous results by Champion, De Pascale, and Prinari.

Provided a new methodological approach using Perron's method.

Abstract

In this paper the existence of an absolute minimizer for a functional \[ F(u,\Omega) = \underset{x \in \Omega}{\text{ess sup}} \, f (x, u(x), Du(x)) \] is proved by using Perron's method. The function is assumed to be quasiconvex and uniformly coercive. This completes the result by Champion, De Pascale and Prinari.