Tuan, NH, Thang, LD and Lesnic, D (2016) A new general filter regularization method for Cauchy problems for elliptic equations with a locally Lipschitz nonlinear source. Journal of Mathematical Analysis and Applications, 434 (2). 1376 - 1393. ISSN 0022-247X
Abstract
Up to now, studies on the semi-linear Cauchy problem for elliptic partial differential equations needed to assume that the source term present in the governing equation is a global Lipschitz function. The current paper is the first investigation to not only the more general but also the more practical case of interest when the source term is only a local Lipschitz function. In such a situation, the methods of solution from the previous studies with a global Lipschitz source term are not directly applicable and therefore, novel ideas and techniques need to be developed to tackle the local Lipschitz nonlinearity. This locally Lipschitz source arises in many applications of great physical interest governed by, for example, the sine-Gordon, Lane–Emden, Allen–Cahn and Liouville equations. The inverse problem is severely ill-posed in the sense of Hadamard by violating the continuous dependence upon the input Cauchy data. Therefore, in order to obtain a stable solution we consider theoretical aspects of regularization of the problem by a new generalized filter method. Under some priori assumptions on the exact solution, we prove and obtain rigorously convergence estimates.
Metadata
Item Type: | Article |
---|---|
Authors/Creators: |
|
Copyright, Publisher and Additional Information: | © 2015 Elsevier. This is an author produced version of a paper accepted for publication in Journal of Mathematical Analysis and Applications. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | Cauchy problem; Nonlinear elliptic equation; Ill-posed problem; Error estimates |
Dates: |
|
Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Applied Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 12 Oct 2015 12:05 |
Last Modified: | 25 Oct 2016 00:42 |
Published Version: | http://dx.doi.org/10.1016/j.jmaa.2015.09.085 |
Status: | Published |
Publisher: | Elsevier |
Identification Number: | 10.1016/j.jmaa.2015.09.085 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:90418 |