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September  2006, 5(3): 415-422. doi: 10.3934/cpaa.2006.5.415

On the existance of minimizers of the variable exponent Dirichlet energy integral

Peter A. Hästö 1,

1. 

Department of Mathematics and Statistics, P.O. Box 68, FI-00014 University of Helsinki, Finland

Received  September 2005 Revised  February 2006 Published  June 2006

In this note we consider the Dirichlet energy integral in the variable exponent case under minimal assumptions on the exponent. First we show that the Dirichlet energy integral always has a minimizer if the boundary values are in $L^\infty$. Second, we give an example which shows that if the so-called "jump-condition", known to be sufficient, is violated, then a minimizer need not exist for unbounded boundary values.
Keywords: p(x)-Laplacian, Variable exponent Sobolev spaces, Dirichlet energy integral..
Mathematics Subject Classification: Primary: 46E35; Secondary: 31C45, 35J6.
Citation: Peter A. Hästö. On the existance of minimizers of the variable exponent Dirichlet energy integral. Communications on Pure and Applied Analysis, 2006, 5 (3) : 415-422. doi: 10.3934/cpaa.2006.5.415
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