# American Institute of Mathematical Sciences

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December  2019, 39(12): 7013-7029. doi: 10.3934/dcds.2019242

## Remarks on some minimization problems associated with BV norms

 1 Rutgers University, Dept. of Math., Hill Center, Busch Campus, 110 Frelinghuysen RD, Piscataway, NJ 08854, USA 2 Dept. of Math. and Dept. of Computer Sc., Technion, 32.000 Haifa, Israel

To Luis Caffarelli, a master of regularity, with esteem and affection

Received  November 2018 Revised  November 2019 Published  June 2019

Fund Project: This research was partially supported by NSF.

The purpose of this paper is twofold. Firstly I present an optimal regularity result for minimizers of a $1D$ convex functional involving the BV-norm, under Neumann boundary condition. This functional is a simplified version of models occuring in Image Processing. Secondly I investigate the existence of minimizers for the same functional under Dirichlet boundary condition. Surprisingly, this turns out to be a delicate issue, which is still widely open.

Citation: Haïm Brezis. Remarks on some minimization problems associated with BV norms. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7013-7029. doi: 10.3934/dcds.2019242
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