A1 Journal article (refereed)
A necessary condition for Sobolev extension domains in higher dimensions (2024)

García-Bravo, M., Rajala, T., & Takanen, J. (2024). A necessary condition for Sobolev extension domains in higher dimensions. Nonlinear Analysis: Theory, Methods and Applications, 240, Article 113446. https://doi.org/10.1016/j.na.2023.113446

JYU authors or editors

Rajala, Tapio
Takanen, Jyrki

Publication details

All authors or editors: García-Bravo, Miguel; Rajala, Tapio; Takanen, Jyrki

Journal or series: Nonlinear Analysis: Theory, Methods and Applications

ISSN: 0362-546X

eISSN: 1873-5215

Publication year: 2024

Publication date: 01/12/2023

Volume: 240

Article number: 113446

Publisher: Elsevier

Publication country: United Kingdom

Publication language: English

DOI: https://doi.org/10.1016/j.na.2023.113446

Publication open access: Not open

Publication channel open access:

Web address of parallel published publication (pre-print): https://arxiv.org/abs/2207.00541

Abstract

We give a necessary condition for a domain to have a bounded extension operator from 𝐿1,𝑝(𝛺) to 𝐿1,𝑝(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 in R3 that is homeomorphic to a ball and has 3-dimensional boundary.

Keywords: functional analysis

Free keywords: Sobolev extension

Fields of science: