A new general filter regularization method for Cauchy problems for elliptic equations with a locally Lipschitz nonlinear source

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:434

Article

Nguyen Huy Tuan¹  Le Duc Thang²  Lesnic, Daniel³ 

[1] Ton Duc Thang Univ, Fac Math & Stat, Appl Anal Res Grp, Ho Chi Minh City, Vietnam

[2] Vietnam Natl Univ, Univ Sci, Dept Math, Ho Chi Minh City, Vietnam

[3] Univ Leeds, Dept Appl Math, Leeds LS2 9JT, W Yorkshire, England

关键词: Cauchy problem;    Nonlinear elliptic equation;    Ill-posed problem;    Error estimates;   

DOI  :  10.1016/j.jmaa.2015.09.085

来源: Elsevier

【 摘 要 】

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. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free