Calculus of Variations and Geometric Measure Theory

J. Koivu - D. Lučić - T. Rajala

Approximation by BV-extension sets via perimeter minimization in metric spaces

created by rajala1 on 05 May 2023
modified on 24 Mar 2024


Accepted Paper

Inserted: 5 may 2023
Last Updated: 24 mar 2024

Journal: International Mathematics Research Notices
Year: 2023
Doi: 10.1093/imrn/rnae048

ArXiv: 2305.02891 PDF


We show that every bounded domain in a metric measure space can be approximated in measure from inside by closed $BV$-extension sets. The extension sets are obtained by minimizing the sum of the perimeter and the measure of the difference between the domain and the set. By earlier results, in PI-spaces the minimizers have open representatives with locally quasiminimal surface. We give an example in a PI-space showing that the open representative of the minimizer need not be a $BV$-extension domain nor locally John.