Regularity under sharp anisotropic general growth conditions

Abstract

We prove boundedness of minimizers of energy-functionals, forinstance of the anisotropic type (1) below, undersharp assumptions onthe exponents $p_{i}$ in terms of $\overline{p}*$: the Sobolev conjugate exponent of $\overline{p}$; i.e.,$\overline{p}*$ = {n\overline{p}}/{n-\overline{p}}, $$1 / \overline{p}$= $\frac{1}{n}\sum_{i=1}^{n}\frac{1}{p_{i}}$. As a consequence, by mean ofregularity results due to Lieberman [21], we obtain thelocal Lipschitz-continuity of minimizers under sharp assumptionson the exponents of anisotropic growth.