IJPAM: Volume 112, No. 1 (2017)

Title

FUNCTIONALS ON BV SPACE WITH CARATHÉODORY
INTEGRANDS USING CONVEX DUALITY

Authors

T. Wunderli
Department of Mathematics and Statistics
The American University of Sharjah
P.O. Box 26666, UAE

Abstract

We define nonlinear functionals $\int_{\Omega }\varphi (x,Du)$ for $u\in BV\left( \Omega \right) $, by using the convex dual $\varphi ^{\ast }(x,q)$ of Carathéodory functions $\varphi (x,p),$ $\varphi :\Omega \times \mathbb{R} ^{N}\rightarrow \mathbb{R} ,$ that for a.e. $x\in \Omega $ are convex and of a linear growth condition in $p.$ Without the usual assumption of continuity in the $x$ variable, we state conditions on $\varphi $ in which $\int_{\Omega }\varphi (x,Du)$ is lower semicontinuous and compact in $L^{1}$, when $\int_{\Omega }\varphi (x,Du)=\int_{\Omega }\varphi (x,\nabla u)\,dx$ for all $u\in W^{1,1}\left( \Omega \right) ,$ and when the formula $\int_{\Omega }\varphi (x,Du)=\int_{\Omega }\varphi (x,\nabla u)\,dx+\int_{\Omega }\psi (x)\vert D^{s}u\vert$ holds for $u\in BV\left( \Omega \right) $. Additionally, we do not use the theory of convex functions of measures nor Reshetnyak continuity as is most commonly done in the literature. Such functionals have important applications to image restoration.

History

Received: October 13, 2016
Revised: November 15, 2016
Published: January 26, 2017

AMS Classification, Key Words

AMS Subject Classification:
Key Words and Phrases: bounded variation, image restoration, convex dual, Carathéodory function

Download Section

Download paper from here.
You will need Adobe Acrobat reader. For more information and free download of the reader, see the Adobe Acrobat website.

Bibliography

1
Andreu-Vaillo, Fuensanta; Caselles, Vincint; Mazón, Jos

How to Cite?

DOI: 10.12732/ijpam.v112i1.13 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2017
Volume: 112
Issue: 1
Pages: 159 - 175


Google Scholar; DOI (International DOI Foundation); WorldCAT.

CC BY This work is licensed under the Creative Commons Attribution International License (CC BY).