Capacitary characterization of variable exponent Sobolev trace spaces

Abstract

Abstract Let Ω ⊂ ℝn be an open set. We give a new characterization of zero trace functions f∈𝒞(Ω¯)∩W01,p(.)(Ω) f \in \mathcal{C}\left( {\bar \Omega } \right) \cap W_0^{1,p\left( . \right)}\left( \Omega \right) . If in addition Ω is bounded, then we give a sufficient condition for which the mapping f↦ℒp(.),fΩ f \mapsto \mathcal{L}_{p\left( . \right),f}^\Omega from a set of real extended functions f : ∂Ω −→ ℝ to the nonlinear harmonic space (Ω,ℋℒp(.)) is injective, where ℒp(.),fΩ \mathcal{L}_{p\left( . \right),f}^\Omega denotes the Perron-Wiener-Brelot solution for the Dirichlet problem: { ℒp(.)u:=-Δp(.)u+ℬ(.,u)=0in Ω;u=fon ∂Ω, \left\{ {\matrix{{{\mathcal{L}_{p\left( . \right)}}u: = - {\Delta _{p\left( . \right)}}u + \mathcal{B}\left( {.,u} \right) = 0} \hfill & {in\,\Omega ;} \hfill \cr {u = f} \hfill & {on\,\partial \Omega ,} \hfill \cr } } \right. where ℬ is a given Carathéodory function satisfies some structural conditions.