The Dirichlet energy integral on intervals in variable exponent Sobolev spaces

P. Harjulehto, P. Hästö and M. Koskenoja

Abstract In this article we consider Dirichlet energy integral minimizers in variable exponent Sobolev spaces defined on intervals of the real line. We illustrate by examples that the minimizing question is interesting even in this case that is trivial in the classical fixed exponent space. We give an explicit formula for the minimizer, and some simple conditions for when it is convex, concave or Lipschitz continuous. The most surprising conclusion is that there does not exist a minimizer even for every smooth exponent.

2000 Mathematics Subject Classification: 46E35, 31C45, 35J65
Keywords: Variable exponent Sobolev space, zero boundary values, Sobolev capacity, Dirichlet energy integral, minimizing problem

Go to Z. Anal. Anwendungen 22, no. 4.