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 $\frac1p-\frac1q\leq \frac1{n-1}$.