The EMS Publishing House is now EMS Press and has its new home at ems.press.

Please find all EMS Press journals and articles on the new platform.

Zeitschrift für Analysis und ihre Anwendungen


Full-Text PDF (216 KB) | Metadata | Table of Contents | ZAA summary
Volume 22, Issue 4, 2003, pp. 911–923
DOI: 10.4171/ZAA/1179

Published online: 2003-12-31

The Dirichlet Energy Integral on Intervals in Variable Exponent Sobolev Spaces

Petteri Harjulehto[1], Peter Hästö[2] and Mika Koskenoja[3]

(1) University of Helsinki, Finland
(2) University of Helsinki, Finland
(3) University of Helsinki, Finland

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.

Keywords: Variable exponent Sobolev space, zero boundary values, Sobolev capacity, Dirichlet energy integral, minimizing problem

Harjulehto Petteri, Hästö Peter, Koskenoja Mika: The Dirichlet Energy Integral on Intervals in Variable Exponent Sobolev Spaces. Z. Anal. Anwend. 22 (2003), 911-923. doi: 10.4171/ZAA/1179