A necessary condition for Sobolev extension domains in higher dimensions

2022/07/01 by Miguel García‐Bravo, García-Bravo, Miguel, Tapio Rajala +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analytic and geometric function theory #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2207.00541

Abstract

We give a necessary condition for a domain to have a bounded extension operator from L^1,p(Ω) to L^1,p(\mathbb Rⁿ) for the range 1 < p < 2. The condition is given in terms of a power of the distance to the boundary of Ω integrated along the measure theoretic boundary of a set of locally finite perimeter and its extension. This generalizes a characterizing curve condition for planar simply connected domains, and a condition for W^1,1-extensions. We use the necessary condition to give a quantitative version of the curve condition. We also construct an example of an extension domain that is homeomorphic to a ball and has n-dimensional boundary.

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