Year: 2019
Author: Haim Brezis
Analysis in Theory and Applications, Vol. 35 (2019), Iss. 4 : pp. 335–354
Abstract
We explore minimization problems of the form
$$\text{Inf} \left\{ \int^1_0 |u'|+ \sum^k_{i=1} |u(a_i) - f_i|^2 + \alpha \int^1_0 |u|^2\right\},$$
where $u$ is a function defined on $(0,1)$, $(a_i)$ are $k$ given points in $(0,1)$, with $k\geq 2$, $(f_i)$ are $k$ given real numbers, and $\alpha \geq0$ is a parameter taken to be $0$ or $1$ for simplicity. The natural functional setting is the Sobolev space $W^{1,1}(0,1)$. When $\alpha=0$ the Inf is achieved in $W^{1,1}(0,1)$. However, when $\alpha =1$, minimizers need not exist in $W^{1,1} (0,1)$. One is led to introduce a relaxed functional defined on the space $BV(0,1)$, whose minimizers always exist and can be viewed as generalized solutions of the original ill-posed problem.
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/ata.OA-0017
Analysis in Theory and Applications, Vol. 35 (2019), Iss. 4 : pp. 335–354
Published online: 2019-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 20
Keywords: Interpolation minimization problems functions of bounded variation relaxed functional.