Uniform Convexity in Variable Exponent Sobolev Spaces

Abstract

We prove the modular convexity of the mixed norm Lp(ℓ2) on the Sobolev space W1,p(Ω) in a domain Ω⊂Rn under the sole assumption that the exponent p(x) is bounded away from 1, i.e., we include the case supx∈Ωp(x)=∞. In particular, the mixed Sobolev norm is uniformly convex if 1<infx∈Ωp(x)≤supx∈Ωp(x)<∞ and W01,p(Ω) is uniformly convex.