Interpolative gap bounds for nonautonomous integrals

TL;DR

This paper introduces a new method for analyzing nonautonomous, nonuniformly elliptic integrals with $(p,q)$-growth, leading to improved higher integrability results for minimizers through interpolation and fractional Sobolev space estimates.

Contribution

It presents a novel approximation technique using mixed local/nonlocal functionals to derive interpolation bounds for $(p,q)$-growth integrals.

Findings

Established a general interpolation property for $(p,q)$-growth integrals.

Achieved improved bounds for the gap $q/p$ to ensure higher integrability.

Developed a new approximation method involving fractional Sobolev spaces.

Abstract

For nonautonomous, nonuniformly elliptic integrals with so-called $(p,q)$-growth conditions, we show a general interpolation property allowing to get basic higher integrability results for H\"older continuous minimizers under improved bounds for the gap $q/p$. For this we introduce a new method, based on approximating the original, local functional, with mixed local/nonlocal ones, and allowing for suitable estimates in fractional Sobolev spaces.