# American Institute of Mathematical Sciences

September  2010, 9(5): 1311-1318. doi: 10.3934/cpaa.2010.9.1311

## On the regularity of minimizers to degenerate functionals

 1 Dipartimento di Matematica e Informatica Università, degli Studi di Salerno Via Ponte don Melillo, 84084 Fisciano (SA), Italy 2 Dipartimento di Statistica e Matematica per la Ricerca Economica Università, “Parthenope ”Via Medina 40, 80131 Napoli, Italy

Received  October 2009 Revised  January 2010 Published  May 2010

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 $x \in \Omega.$

Citation: P. Di Gironimo, L. D’Onofrio. On the regularity of minimizers to degenerate functionals. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1311-1318. doi: 10.3934/cpaa.2010.9.1311
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