Calculus of Variations and Geometric Measure Theory
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G. Cupini - P. Marcellini - E. Mascolo

Regularity under sharp anisotropic general growth conditions

created by cupini on 18 Nov 2008
modified on 26 Jan 2009

[BibTeX]

Published Paper

Inserted: 18 nov 2008
Last Updated: 26 jan 2009

Journal: Discrete Contin. Dyn. Syst. Ser. B
Volume: 11
Number: 1
Pages: 67-86
Year: 2009

Abstract:

We prove boundedness of minimizers of energy-functionals, for instance of the anisotropic type (\ref{1}) below, \begin{equation} \mathcal{F}(u)=\int{\Omega }\sum{i=1}{n}
u
{x{i}}(x)
{p{i}\left( x\right) }\,dx \label{1} \end{equation} under \textit{sharp} assumptions on the exponents $p_{i}$ in terms of ${\overline{p}}^{\ast }$: the \textit{% Sobolev conjugate exponent} of ${\overline{p}}$; i.e., ${\overline{p}}^{\ast }=\frac{n\overline{p}}{n-\overline{p}}$, $\frac{1}{\overline{p}}=\frac{1}{n}% \sum_{i=1}^{n}\frac{1}{p_{i}}$. As a consequence, by mean of regularity results due to Lieberman \cite{lie}, we obtain the local Lipschitz-continuity of minimizers under sharp assumptions on the exponents of anisotropic growth.

Keywords: Minimizers, $L^{\infty }-$regularity, gradient estimates, anisotropic growth conditions


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