Remarks on some minimization problems associated with BV norms

Haïm Brezis

doi:10.3934/dcds.2019242

Remarks on some minimization problems associated with BV norms

Haïm Brezis

School of Arts and Sciences, Mathematics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

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.

Original language	English (US)
Pages (from-to)	7013-7029
Number of pages	17
Journal	Discrete and Continuous Dynamical Systems- Series A
Volume	39
Issue number	12
DOIs	https://doi.org/10.3934/dcds.2019242
State	Published - 2019

All Science Journal Classification (ASJC) codes

Analysis
Discrete Mathematics and Combinatorics
Applied Mathematics

Keywords

Bounded variation
Dirichlet boundary condition
Neumann boundary condition
Regularity of minimizers

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10.3934/dcds.2019242

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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.",

keywords = "Bounded variation, Dirichlet boundary condition, Neumann boundary condition, Regularity of minimizers",

author = "Ha{\"i}m Brezis",

note = "Funding Information: 2010 Mathematics Subject Classification. Primary: 26A45, 35J20; Secondary: 35J25. Key words and phrases. Bounded variation, regularity of minimizers, Dirichlet boundary condition, Neumann boundary condition. This research was partially supported by NSF. Funding Information: This research was partially supported by NSF. Publisher Copyright: {\textcopyright} 2019 American Institute of Mathematical Sciences. All rights reserved.",

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language = "English (US)",

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N1 - Funding Information: 2010 Mathematics Subject Classification. Primary: 26A45, 35J20; Secondary: 35J25. Key words and phrases. Bounded variation, regularity of minimizers, Dirichlet boundary condition, Neumann boundary condition. This research was partially supported by NSF. Funding Information: This research was partially supported by NSF. Publisher Copyright: © 2019 American Institute of Mathematical Sciences. All rights reserved.

PY - 2019

Y1 - 2019

N2 - 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.

AB - 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.

KW - Bounded variation

KW - Dirichlet boundary condition

KW - Neumann boundary condition

KW - Regularity of minimizers

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JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

IS - 12

ER -

Remarks on some minimization problems associated with BV norms

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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