On approximate solutions to one class of non-linear Dirichlet elliptic boundary value problems

Vìsnik Dnìpropetrovsʹkogo Unìversitetu: Serìâ Modelûvannâ

卷:24

A. O. Putchenko¹  P. I. Kogut¹ 

[1] Oles Honchar Dnipropetrovsk National University;

关键词: existence result;    elliptic equations;    fictitious control;    perturbation approach;   

DOI  :  10.15421/141603

来源: DOAJ

【 摘 要 】

We discuss the existence of weak solutions to one class of Dirichlet boundary value problems (BVP) for non-linear elliptic equation. Because of the specic of nonlinearity, we cannot a priori expect to have a solution in the standard functional space. Instead of this we show that the original BVP admits the so-called approximate weak solution. To do so, we introduce a special family of perturbed optimal control problems (OCPs) where the class of ctitious controls are closely related with the properties of nonlinearity in right-hand side of the elliptic equation. The main question we discuss in this paper is about solvability of perturbed OCPs, uniqueness of their solutions, and asymptotic properties of optimal pairs as the perturbation parameter " > 0 tends to zero. As a result, we derive the sucient conditions of the existence of weak solutions to the given class of nonlinear Dirichlet BVP and give a way for the approximation of such solutions.

【 授权许可】

Unknown