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

2022/12/20 by Ginster, Janusz, Gladbach, Peter · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2212.10199

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.

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