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.
@article{urn:eudml:doc:44531,
title = {A capacity approach to the Poincar\'e inequality and Sobolev imbeddings in variable exponent Sobolev spaces.},
journal = {Revista Matem\'atica de la Universidad Complutense de Madrid},
volume = {17},
year = {2004},
pages = {129-146},
zbl = {1072.46021},
mrnumber = {MR2063945},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:44531}
}
Harjulehto, Petteri; Hästö, Peter. A capacity approach to the Poincaré inequality and Sobolev imbeddings in variable exponent Sobolev spaces.. Revista Matemática de la Universidad Complutense de Madrid, Tome 17 (2004) pp. 129-146. http://gdmltest.u-ga.fr/item/urn:eudml:doc:44531/