### 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.