On an inverse boundary value problem of a nonlinear elliptic equation in three dimensions

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:426

Article

Nguyen Huy Tuan¹  Le Duc Thang²  Vo Anh Khoa^1,4  Thanh Tran³ 

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

[2] Ho Chi Minh City Ind & Trade Coll, Fac Basic Sci, Ho Chi Minh City, Vietnam

[3] Univ New S Wales, Sch Math & Stat, Sydney, NSW 2052, Australia

[4] Gran Sasso Sci Inst, Math & Comp Sci Div, I-67100 Laquila, Italy

关键词: Nonlinear elliptic equation;    Ill-posed problem;    Regularization;    Truncation method;   

DOI  :  10.1016/j.jmaa.2014.12.047

来源: Elsevier

【 摘 要 】

This work considers an inverse boundary value problem for a 3D nonlinear elliptic partial differential equation in a bounded domain. In general, the problem is severely ill-posed. The formal solution can be written as a hyperbolic cosine function in terms of the 2D elliptic operator via its eigenfunction expansion, and it is shown that the solution is stabilized or regularized if the large eigenvalues are cut off. In a theoretical framework, a truncation approach is developed to approximate the solution of the ill-posed problem in a regularization manner. Under some assumptions on regularity of the exact solution, we obtain several explicit error estimates including an error estimate of Holder type. A local Lipschitz case of source term for this nonlinear problem is obtained. For numerical illustration, two examples on the elliptic sine-Gordon and elliptic Allen Cahn equations are constructed to demonstrate the feasibility and efficiency of the proposed methods. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free