On the ill-posed problem for the Poisson equation

Abstract

A boundary value problem in a two-dimensional rectangular region for the Poisson equation is studied in the paper. The original ill-posed boundary value problem is transformed to the optimal control problem. The paper gives a brief overview of the problem under study, defines the formulation of the original boundary value problem and optimization problems, proves the existence of a solution to the regularized optimization problem, determines the formulation of the adjoint boundary value problem, studies the optimality conditions, and presents the application of the variable separation method. The necessary and sufficient conditions of optimality in terms of the conjugate boundary value problem are established in the paper, and a strong criterion for the solvability of the ill-posed boundary value problem is obtained. Boundary value problems for the Poisson equation arise in many sections of physics, mechanics, and otherapplied sciences. So, the stress function in the torsion problem of elastic rods is the solution of the Dirichlet problem, and the height of the liquid rise in the cylindrical capillary is the solution of the Neumann problem. But in many cases practitioners are interested in ill-posed problems for the Poisson equation and their solvability, which determines the relevance of the problem studied in the article.