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

TL;DR

This paper establishes Lipschitz regularity for bounded minimizers of functionals with nonstandard (p,q)-growth, improving existing restrictions and providing sharp results in certain dimension ranges.

Contribution

The authors prove Lipschitz estimates for minimizers under a less restrictive (p,q)-growth condition, extending the regularity theory for such functionals.

Findings

Lipschitz estimates hold under q < p+2 for p ≥ 2

Improves previous restrictions when p ≤ N-1

Results are sharp for N > p(2+p)/2 + 1

Abstract

We obtain Lipschitz estimates for bounded minimizers of functionals with nonstandard $(p,q)$-growth satisfying the dimension-independent restriction $q<p+2$ with $p \geq 2$. This relation improves existing restrictions when $p \leq N-1$, moreover our result is sharp in the range $N > \frac{p(2+p)}{2} + 1$. The standard Lipschitz regularity takes the form $W^{1,\infty}_{\text{loc}} - W^{1,p}_{\text{loc}}$, whereas we obtain $W^{1,\infty}_{\text{loc}} - L^{\infty}_{\text{loc}}$ regularity estimate and then make use of existing sharp $L^{\infty}_{\text{loc}}$ bounds to obtain the required conclusion.