$% P. Harjulehto and P. Hästö, A capacity approach to the Poincaré inequality and Sobolev imbeddings in variable exponent Sobolev spaces, Rev. Mat. Complut. 17 (2004), 129–146.%$

A Capacity Approach to the Poincaré Inequality and Sobolev Imbeddings in Variable Exponent Sobolev Spaces

Petteri HARJULEHTO and Peter HÄSTÖ

Department of Mathematics

P.O. Box 4 (Yliopistonkatu 5)

FIN-00014 University of Helsinki — Finland

petteri.harjulehto@helsinki.fi

peter.hasto@helsinki.fi

P.O. Box 4 (Yliopistonkatu 5)

FIN-00014 University of Helsinki — Finland

petteri.harjulehto@helsinki.fi

peter.hasto@helsinki.fi

Accepted: September 10, 2003

ABSTRACT

We study the Poincaré inequality in Sobolev spaces with variable exponent. Under a rather mild and sharp condition on the exponent p we show that the inequality holds. This condition is satisfied e.g. if the exponent p is continuous in the closure of a convex domain. We also give an essentially sharp condition for the exponent p as to when there exists an imbedding from the Sobolev space to the space of bounded functions.

**Key words: **Sobolev spaces, variable exponent, Poincaré inequality, Sobolev imbedding,
continuity.**
2000 Mathematics Subject Classification:** 46E35.