An anisotropic Poincar\'e inequality in $GSBV^p$ and the limit of strongly anisotropic Mumford-Shah functionals

TL;DR

This paper establishes an anisotropic Poincaré inequality in $GSBV^p$ functions, demonstrating their proximity to one-dimensional functions outside small exceptional sets, and applies this to analyze the limit of anisotropic Mumford-Shah functionals via $ ext{Gamma}$-convergence.

Contribution

It introduces a new anisotropic Poincaré inequality in $GSBV^p$ and proves $ ext{Gamma}$-convergence of anisotropic Mumford-Shah functionals to a one-dimensional model.

Findings

Functions with small variation in two directions are close to one-dimensional functions.

Provides bounds on volume and perimeter of exceptional sets.

Establishes $ ext{Gamma}$-convergence of anisotropic Mumford-Shah functionals.

Abstract

We show that functions in $GSBV^p$ in three-dimensional space with small variation in $2$ of $3$ directions are close to a function of one variable outside an exceptional set. Bounds on the volume and the perimeter in these two directions of the exceptional sets are provided. As a key tool we prove an approximation result for such functions by functions in $W^{1,p}$. For this we present a two-dimensional countable ball construction that allows to carefully remove the jumps of the function. As a direct application, we show $\Gamma$-convergence of an anisotropic three-dimensional Mumford-Shah model to a one-dimensional model.