A Poincaré inequality in a Sobolev space with a variable exponent

Let Ω be a domain in ^N. It is shown that a generalized Poincaré inequality holds in cones contained in the Sobolev space W^1,p(·)(ω), where p(·):[1, ∞[ is a variable exponent. This inequality is itself a corollary to a more general result about equivalent norms over such cones. The approach in this paper avoids the difficulty arising from the possible lack of density of the space D(Ω) in the space {v ∈ W^1,p(·)(Ω); tr v = 0 on ∂Ω}. Two applications are also discussed. © 2011 Editorial Office of CAM (Fudan University) and Springer Berlin Heidelberg.