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 ¯ ) ∩ W 0 1 , 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 ↦ ℒ p ( . ) , f Ω 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,ℋℒ p (.) ) is injective, where ℒ p ( . ) , f Ω \mathcal{L}_{p\left( . \right),f}^\Omega denotes the Perron-Wiener-Brelot solution for the Dirichlet problem: { ℒ p ( . ) u : = - Δ p ( . ) u + ℬ ( . , u ) = 0 i n   Ω ; u = f o n   ∂ 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.