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

Peter A. Hästö

doi:10.3934/cpaa.2006.5.415

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2006, Volume 5, Issue 3: 415-422. Doi: 10.3934/cpaa.2006.5.415

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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

Received: August 31, 2005

Revised: January 31, 2006

Published: August 2006

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Abstract

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.

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Mathematics Subject Classification: Primary: 46E35; Secondary: 31C45, 35J65.

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