Calculus of Variations and Geometric Measure Theory

L. MalĂ˝ - N. Shanmugalingam - M. Snipes

Trace and extension theorems for functions of bounded variation

created by shanmugal on 27 Apr 2016
modified on 28 Nov 2016


Accepted: 2016

Inserted: 27 apr 2016
Last Updated: 28 nov 2016

Journal: Annali SNS.
Year: 2016
Doi: 10.2422/2036-2145.201511_007


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 $1$-Poincar\'e inequality. We also construct a bounded linear extension from a Besov class of functions on $\partial\Omega$ to $BV(\Omega)$.