Self-improving property for certain degenerate functionals with generalized Orlicz growth

TL;DR

This paper studies the higher integrability of gradients for local quasiminimizers of certain variational integrals with generalized Orlicz growth, under weighted conditions, and proves the existence of minimizers.

Contribution

It establishes a self-improving property and higher integrability results for variational integrals with generalized Orlicz growth in a weighted setting, and proves the existence of minimizers.

Findings

Gradient of quasiminimizers has local higher integrability.

Existence of minimizers for the functional is proven.

Results apply under Muckenhoupt weight conditions.

Abstract

We investigate a self-improving property of variational integrals in a weighted framework under generalized Orlicz growth conditions. Assuming that the weight belongs to an appropriate Muckenhoupt class and the growth function satisfies standard structural conditions, we prove that the gradient of any local quasiminimizer has local higher integrability. In addition, we establish the existence of minimizers for the associated functional.