An anisotropic Poincaré inequality in GSBV^p and the limit of strongly anisotropic Mumford–Shah functionals

¹ Weierstrass Institute, Anton-Wilhelm-Amo-Strasse 39, 10117 Berlin, Germany
² Peter Gladbach Institut f¨ur Angewandte Mathematik Rheinische Friedrich-Wilhelms - Universität Bonn Endenicher Allee 60 53115 Bonn, Germany

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 2 January 2023
Accepted: 2 January 2025

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 Γ-convergence of an anisotropic three-dimensional Mumford-Shah model to a one-dimensional model.