Communications on Pure and Applied Analysis (CPAA)

On the regularity of minimizers to degenerate functionals

Pages: 1311 - 1318, Volume 9, Issue 5, September 2010

doi:10.3934/cpaa.2010.9.1311       Abstract        Full Text (168.6K)       Related Articles

P. Di Gironimo - Dipartimento di Matematica e Informatica Università, degli Studi di Salerno Via Ponte don Melillo, 84084 Fisciano (SA), Italy (email)
L. D’Onofrio - Dipartimento di Statistica e Matematica per la Ricerca Economica Università, “Parthenope ”Via Medina 40, 80131 Napoli, Italy (email)

Abstract: In this paper, we prove a higher integrability result for the gradient of a minimizer of a functional of the type

$I(\Omega , u)=\int_{\Omega}\sum_{i,j} a_{i,j} D_i u D_jv dx$

whose coefficient matrix $A(x)= ^tA(x)$ satisfies the anisotropic bounds

$\frac{|\xi |^2}{K(x)}\leq < A(x) \xi, \xi > \leq K(x) |\xi |^2\quad \forall \xi \in R^n,$ for a.e. $x\in \Omega,$

where $ K:\Omega \subset R^n \rightarrow [1,+\infty),$ a locally integrable function in $\Omega$, belongs to $A_2 \cap G_n$ and has a majorant $Q(x)\geq K(x)$ of finite mean,

limsup$_{R \rightarrow 0} \int_{B_R(x)} Q(y)dy < \infty $ at every point $x \in \Omega. $

Keywords:  Partial differential equations, weights, finite distortion.
Mathematics Subject Classification:  42B25, 42B35.

Received: October 2009;      Revised: January 2010;      Published: May 2010.