A Poincaré Inequality in a Sobolev Space with a Variable Exponent

Let Ω be a domain in RN.It is shown that a generalized Poincaré inequality holds in cones contained in the Sobolev space W1,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 ∈ W1,p(·)(Ω);trv = 0 on ■Ω}.Two applications are also discussed.

Let Ω be a domain in RN. It is shown that a generalized Poincaré inequality holds in cones contained in the Sobolev space Wl,P（ ）（Ω）, where p（.） ： Ω → [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 ∈ Wl,P（ ）（Ω）; tr v = 0 on δΩ}. Two applications are also discussed.