Growth conditions and regularity, an optimal local boundedness result

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 $\frac{1}{p}-\frac{1}{q}\le \frac{1}{n-1}$.

Corrected statement and proof of Theorem 2 regarding the critical case $p=n$