Borderline Lipschitz regularity for bounded minimizers of functionals with (p,q)-growth

TL;DR

This paper establishes local Lipschitz regularity for bounded minimizers of functionals with nonstandard (p,q)-growth, extending previous results under weaker conditions and identifying sharp parameter ranges.

Contribution

It extends the regularity results for minimizers with (p,q)-growth to weaker hypotheses, providing sharp conditions for Lipschitz continuity.

Findings

Proves local Lipschitz regularity for bounded minimizers.

Identifies sharp parameter restrictions for (p,q)-growth functionals.

Extends previous work by Beck-Mingione to broader conditions.

Abstract

We prove local Lipschitz regularity for bounded minimizers of functionals with nonstandard $p,q$-growth with the source term in the Lorentz space $L(N,1)$ under the restriction $q<p+1+p\,\min\left\{\frac 1N,\frac{2(p-1)}{Np-2p+2}\right\}$. This extends the recent work by Beck-Mingione to bounded minimizers under weaker hypothesis and is sharp for some special ranges of $p$, $q$ and $N$.