The Dirichlet Energy Integral on Intervals in Variable Exponent Sobolev Spaces

Abstract

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.