A Poincare Inequality in a Sobolev Space with a Variable Exponent

Let Omega be a domain in R-N. It is shown that a generalized Poincare inequality holds in cones contained in the Sobolev space W-1,W-p(.)(Omega), where p(.) : (Omega) over bar -> [1,infinity[ 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(Omega) in the space {v is an element of W-1,W-p(.)(Omega); tr v = 0 on partial derivative Omega}. Two applications are also discussed.