Regularity under sharp anisotropic general growth conditions

Giovanni Cupini; Paolo Marcellini; Elvira Mascolo

doi:10.3934/dcdsb.2009.11.67

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2009, Volume 11, Issue 1: 67-86. Doi: 10.3934/dcdsb.2009.11.67

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Regularity under sharp anisotropic general growth conditions

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

Received: November 30, 2007

Revised: April 30, 2008

Published: June 2009

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Abstract

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.

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Mathematics Subject Classification: Primary: 49N60; Secondary: 35J70.

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