A Capacity Approach to the Poincaré Inequality and Sobolev Imbeddings in Variable Exponent Sobolev Spaces
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,
2000 Mathematics Subject Classification: 46E35.