Estimates of the derivatives of minimizersof a special class of variational integrals

Maria Alessandra Ragusa; Atsushi Tachikawa

doi:10.3934/dcds.2011.31.1411

Article Contents

2011, Volume 31, Issue 4: 1411-1425. Doi: 10.3934/dcds.2011.31.1411

This issue Previous Article Quasilinear divergence form parabolic equations in Reifenberg flat domains Next Article Gamma-convergence of gradient flows on Hilbert and metric spaces and applications

Estimates of the derivatives of minimizers of a special class of variational integrals

Maria Alessandra Ragusa^1, and
Atsushi Tachikawa^2,

1.
Dipartimento di Matematica, Università di Catania, Università di Catania, Viale A. Doria, 6, 95125 Catania
2.
Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, Noda, Chiba, 278-8510

Received: May 31, 2009

Revised: August 31, 2010

Published: September 2011

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Abstract

The note concerns on some estimates in Morrey Spaces for the derivatives of local minimizers of variational integrals of the form $$\int_\Omega F (x,u,Du) dx $$ where the integrand has the following special form $$ F(x,u,Du)\, =\, A(x,u, g^{\alpha\beta}(x) h_{ij}(u) \frac{\partial u^i}{\partial x^\alpha} \frac{\partial u^i }{\partial x^\beta}), $$ where $(g^{\alpha\beta})$ and $(h_{ij})$ symmetric positive definite matrices. We are not assuming the continuity of $A$ and $g$ with respect to $x$. We suppose that $A(\cdot, u,t)/(1+t)$ and $g(\cdot)$ are in the class $L^\infty\cap VMO$.

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Mathematics Subject Classification: Primary: 49N60, 35J50, 35R05; Secondary: 46E30, 35B65.

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