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Discrete and Continuous Dynamical Systems - Series B (DCDS-B)
 

Regularity under sharp anisotropic general growth conditions

Pages: 67 - 86, Volume 11, Issue 1, January 2009      doi:10.3934/dcdsb.2009.11.67

 
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Giovanni Cupini - Dipartimento di Matematica "U. Dini", Università di Firenze, Viale Morgagni 67/A, 50134 - Firenze, Italy (email)
Paolo Marcellini - Dipartimento di Matematica "U. Dini", Università di Firenze, Viale Morgagni 67/A, 50134 - Firenze, Italy (email)
Elvira Mascolo - Dipartimento di Matematica "U. Dini", Università di Firenze, Viale Morgagni 67/A, 50134 - Firenze, Italy (email)

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.

Keywords:  $L^{\infty }-$regularity, gradient estimates, minimizers, anisotropic growth conditions, $p-q$ growth conditions
Mathematics Subject Classification:  Primary: 49N60; Secondary: 35J70

Received: December 2007;      Revised: May 2008;      Available Online: November 2008.