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

Author information: Dr. Nguyen Huy Tuan, Head of Applied Analysis Research Group of Ton Duc Thang University (AA-RG).

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

Journal information: The paper was published in the ISI journal Journal of Mathematical Analysis and Applications (its impact factor is 1.120 and H-index is 102).

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.

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