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arXiv · 2503.15716

Closed BV-extension and $W^{1,1}$-extension sets

Emanuele Caputo·Jesse Koivu·Danka Lučić·Tapio Rajala
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Abstract

This paper studies the relations between extendability of different classes of Sobolev $W^{1,1}$ and $BV$ functions from closed sets in general metric measure spaces. Under the assumption that the metric measure space satisfies a weak $(1,1)$-Poincar\'e inequality and measure doubling, we prove further properties for the extension sets. In the case of the Euclidean plane, we show that compact finitely connected $BV$-extension sets are always also $W^{1,1}$-extension sets. This is shown via a local quasiconvexity result for the complement of the extension set.

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Emanuele Caputo, Jesse Koivu, Danka Lučić, Tapio Rajala. 2025-03-19. Closed BV-extension and $W^{1,1}$-extension sets. https://arxiv.org/abs/2503.15716

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