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

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

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