Remarks on some minimization problems associated with BV norms

Haïm Brezis

doi:10.3934/dcds.2019242

Article Contents

2019, Volume 39, Issue 12: 7013-7029. Doi: 10.3934/dcds.2019242

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Remarks on some minimization problems associated with BV norms

Haïm Brezis^1,2,

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

Received: October 31, 2018

Revised: October 31, 2019

Published: September 2019

This research was partially supported by NSF.

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Abstract

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.

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Mathematics Subject Classification: Primary: 26A45, 35J20; Secondary: 35J25.

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