Compactness of special functions of bounded higher variation

TL;DR

This paper introduces new function spaces, GB_nV and GSB_nV, which generalize existing spaces to better describe singularities of higher codimension, and establishes key compactness and lower semicontinuity properties.

Contribution

The paper defines the spaces GB_nV and GSB_nV of generalized functions of bounded higher variation, extending prior models to include singularities of codimension n.

Findings

Established compactness of sublevel sets in GSB_nV.

Proved lower semicontinuity of Mumford-Shah type functionals.

Generalized the concept of bounded variation to higher codimension singularities.

Abstract

Given an open set \Omega\subset\R^m and n>1, we introduce the new spaces GB_nV(\Omega) of Generalized functions of bounded higher variation and GSB_nV(\Omega) of Generalized special functions of bounded higher variation that generalize, respectively, the space B_nV introduced by Jerrard and Soner and the corresponding SB_nV space studied by De Lellis. In this class of spaces, which allow the description of singularities of codimension n, the distributional jacobian Ju need not have finite mass: roughly speaking, finiteness of mass is not required for the (m-n)-dimensional part of Ju, but only finiteness of size. In the space GSB_nV we are able to provide compactness of sublevel sets and lower semicontinuity of Mumford-Shah type functionals, in the same spirit of the codimension 1 theory.