Duality of capacities and Sobolev extendability in the plane

Yi Ru-Ya Zhang

doi:10.1007/s40627-021-00063-2

Abstract

We reveal relations between the duality of capacities and the duality between Sobolev extendability of Jordan domains in the plane, and explain how to read the curve conditions involved in the Sobolev extendability of Jordan domains via the duality of capacities. Finally as an application, we give an alternative proof of the necessary condition for a Jordan planar domain to be W^{1,,q}-extension domain when 2<q<infty.

Highlights

Let Ω be a Jordan domain in the plane, that is, Ω is the bounded connected component of R2 ⧵ 0 for a Jordan curve 0
We say that Ω is a W1, p-extension domain if it admits an extension operator E ∶ W1,p(Ω) → W1,p(R2) such that, there exists a constant C ≥ 1 so that for every u ∈ W1,p(Ω) we have ||Eu||W1,p(R2) ≤ C||u||W1,p(Ω) and Eu|Ω = u
Theorem 1.2 ([1, 19]) Let Ω ⊂ Ĉ be a Jordan domain enclosed by four arcs 1, 2, 3 and 4 counterclockwise

Summary

Introduction

Let Ω be a Jordan domain in the plane, that is, Ω is the bounded connected component of R2 ⧵ 0 for a Jordan curve 0. We define the Sobolev space W1,p(Ω) , 1 ≤ p ≤ ∞, as W1,p(Ω) = u ∈ L1loc (Ω) ∶ ∇u ∈ Lp(Ω, R2) , where ∇u denotes the distributional gradient of u. Theorem 1.1 Let Ω ⊂ C be a Jordan domain. It is a W1, p-extension domain if and only if Ω ∶= C ⧵ Ω is a W1, q -extension domain for 1 < p < ∞ and q = p−p1. Recall that for a given pair of continua E, F ⊂ Ω ⊂ R2 and 1 < p < ∞ , one defines the p -capacity between E and F in Ω as. Theorem 1.2 ([1, 19]) Let Ω ⊂ Ĉ be a Jordan domain enclosed by four arcs 1 , 2 , 3 and 4 counterclockwise. To see how the duality in Theorem 1.1 comes from (1.1), suppose Ω is a W1, q-extension Jordan domain with 2 < q < ∞ and write

Page 2 of 6

Prerequisites

Page 4 of 6

Compliance with ethical standards