Trace and extension theorems for functions of bounded variation

In this paper we show that every \(L^1\)-integrable function on \(\partial\Omega\) can be obtained as the trace of a function of bounded variation in \(\Omega\) whenever \(\Omega\) is a domain with regular boundary \(\partial\Omega\) in a doubling metric measure space. In particular, the trace class...

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Bibliographic Details
Published in:arXiv.org 2016-07
Main Authors: Malý, Lukáš, Nageswari Shanmugalingam, Snipes, Marie
Format: Article
Language:English
Subjects:
Mathematical functions
Online Access:Get full text
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Summary:In this paper we show that every \(L^1\)-integrable function on \(\partial\Omega\) can be obtained as the trace of a function of bounded variation in \(\Omega\) whenever \(\Omega\) is a domain with regular boundary \(\partial\Omega\) in a doubling metric measure space. In particular, the trace class of \(BV(\Omega)\) is \(L^1(\partial\Omega)\) provided that \(\Omega\) supports a 1-Poincaré inequality. We also construct a bounded linear extension from a Besov class of functions on \(\partial\Omega\) to \(BV(\Omega)\).
ISSN:2331-8422