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.

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