Growth conditions and regularity, an optimal local boundedness result

TL;DR

This paper establishes an optimal local boundedness result for scalar minimizers of integral functionals with $(p,q)$-growth conditions, extending regularity theory in the calculus of variations.

Contribution

It proves local boundedness of minimizers under the optimal relation between growth exponents, improving understanding of regularity for non-standard growth conditions.

Findings

Proves local boundedness of minimizers under optimal $(p,q)$-growth conditions.

Identifies the precise relation $rac1p-rac1q\leq rac1{n-1}$ as critical for regularity.

Extends regularity results to broader classes of integral functionals.

Abstract

We prove local boundedness of local minimizers of scalar integral functionals $\int_\Omega f(x,\nabla u(x))\,dx$, $\Omega\subset\mathbb R^n$ where the integrand satisfies $(p,q)$-growth of the form \begin{equation*} |z|^p\lesssim f(x,z)\lesssim |z|^q+1 \end{equation*} under the optimal relation $\frac1p-\frac1q\leq \frac1{n-1}$.