arXiv Open Access 2018

A uniqueness result for functions with zero fine gradient on quasiconnected and finely connected sets

Anders Björn Jana Björn

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Abstrak

We show that every Sobolev function in $W^{1,p}_{\textrm{loc}}(U)$ on a $p$-quasiopen set $U \subset {\bf R}^n$ with a.e.-vanishing $p$-fine gradient is a.e.-constant if and only if $U$ is $p$-quasiconnected. To prove this we use the theory of Newtonian Sobolev spaces on metric measure spaces, and obtain the corresponding equivalence also for complete metric spaces equipped with a doubling measure supporting a $p$-Poincaré inequality. On unweighted ${\bf R}^n$, we also obtain the corresponding result for $p$-finely open sets in terms of $p$-fine connectedness, using a deep result by Latvala.

Topik & Kata Kunci

math.AP math.FA

Penulis (2)

Anders Björn

Jana Björn

Format Sitasi

APA MLA BibTeX

Björn, A., Björn, J. (2018). A uniqueness result for functions with zero fine gradient on quasiconnected and finely connected sets. https://arxiv.org/abs/1802.06031

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Informasi Jurnal

Tahun Terbit: 2018
Bahasa: en
Sumber Database: arXiv
Akses: Open Access ✓