Direct and inverse problems for the Poisson equation with equality of flows on a part of the boundary

ABSTRACT

In the paper we consider a stationary diffusion problem described by the Poisson equation. The problem is considered in a model domain, chosen as a half disk. Classical Dirichlet boundary conditions are set on the arc of the circle. New nonlocal boundary conditions are set on the bottom base. The first condition means the equality of flows through opposite radii, and the second condition is the proportionality of distribution densities on these radii with a variable coefficient of proportionality. Uniqueness and existence of the classical solution to the problem are proved. An inverse problem for the solution to the Poisson equation and its right-hand part depending only on an angular variable are considered. As an additional condition we use the boundary overdetermination. Inverse problems to the Dirichlet and Neumann problems, and to problems with nonlocal conditions of the equality of flows through the opposite radii are considered. The well-posedness of the formulated inverse problems is proved.

COMMUNICATED BY:

• A. Soldatov

KEYWORDS:

• Poisson equation
• Dirichlet problem
• Neumann problem
• nonlocal boundary conditions
• inverse problem
• boundary overdetermination

AMS SUBJECT CLASSIFICATIONS:

• 35J05
• 35J25
• 35R30

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Makhmud A. Sadybekov  http://orcid.org/0000-0001-8450-8191

Additional information

Funding

This paper was published under project AP05133271 of the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan.