An anisotropic Poincar\'e inequality in $GSBV^p$ and the limit of strongly anisotropic Mumford-Shah functionals
| Autor: | Ginster, Janusz, Gladbach, Peter |
|---|---|
| Rok vydání: | 2022 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. |
| Databáze: | arXiv |
| Externí odkaz: |
|