A modified integral equation method of the nonlinear elliptic equation with globally and locally Lipschitz source

TL;DR

This paper introduces a modified integral equation method to regularize and solve a severely ill-posed nonlinear elliptic PDE Cauchy problem in Hilbert spaces, with proven convergence and numerical validation.

Contribution

It develops a novel regularization technique for nonlinear elliptic PDEs with Lipschitz sources, extending previous methods and providing convergence analysis and numerical evidence.

Findings

Method effectively regularizes the nonlinear problem.

Convergence estimates are established under a priori assumptions.

Numerical tests confirm the method's feasibility and effectiveness.

Abstract

The paper is devoted to investigating a Cauchy problem for nonlinear elliptic PDEs in the abstract Hilbert space. The problem is hardly solved by computation since it is severely ill-posed in the sense of Hadamard. We shall use a modified integral equation method to regularize the nonlinear problem with globally and locally Lipschitz source terms. Convergence estimates are established under priori assumptions on exact solution. A numerical test is provided to illustrate that the proposed method is feasible and effective. These results extend some earlier works on a Cauchy problem for elliptic equations.