FRÉCHET DIFFERENTIABILITY OF THE NORM IN A SOBOLEV SPACE WITH A VARIABLE EXPONENT

Abstract

Let Ω be a domain in ℝN, let [Formula: see text] be such that p(x) > 1 for all [Formula: see text], let W1,p(⋅)(Ω) be the Sobolev space with variable exponent p(⋅), let Γ0be a dΓ-measurable subset of Γ = ∂Ω that satisfies dΓ-meas Γ0> 0, and let UΓ0= {u ∈ W1,p(⋅)(Ω); tr u = 0 on Γ0}.It is shown that the map u ∈ UΓ0↦ ‖u‖0,p(⋅), ∇= ‖|∇u|‖0,p(⋅)is a Fréchet-differentiable norm on UΓ0, and a formula expressing the Fréchet derivative of this norm at any nonzero u ∈ UΓ0is given.We also show that, if p(x) ≥ 2 for all [Formula: see text], (UΓ0, ‖u‖0,p(⋅), ∇) is uniformly convex.Using properties of duality mappings defined on Banach spaces having a Fréchet-differentiable norm, we give the explicit form of continuous linear functionals on (UΓ0, ‖u‖0,p(⋅), ∇). It is also shown that the space UΓ0and its dual have the same Krein–Krasnoselski–Milman dimension.