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

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 Hölder 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.