# American Institute of Mathematical Sciences

January  2009, 11(1): 67-86. doi: 10.3934/dcdsb.2009.11.67

## Regularity under sharp anisotropic general growth conditions

 1 Dipartimento di Matematica "U. Dini", Università di Firenze, Viale Morgagni 67/A, 50134 - Firenze, Italy, Italy

Received  December 2007 Revised  May 2008 Published  November 2008

We prove boundedness of minimizers of energy-functionals, for instance of the anisotropic type (1) below, under sharp assumptions on the 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}}$. As a consequence, by mean of regularity results due to Lieberman [21], we obtain the local Lipschitz-continuity of minimizers under sharp assumptions on the exponents of anisotropic growth. Citation: Giovanni Cupini, Paolo Marcellini, Elvira Mascolo. Regularity under sharp anisotropic general growth conditions. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 67-86. doi: 10.3934/dcdsb.2009.11.67  [1] Paolo Marcellini. Regularity under general and$ p,q- $growth conditions. 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