Problem for the diffusion equation outside cuts on the plane with the Dirichlet condition and an oblique derivative condition on opposite sides of the cuts

• Published in: Differential equations Vol. 48; no. 9; pp. 1197 - 1211
• Main Authors: Krutitskii, P. A., Prozorov, K. V.
• Format: Journal Article
• Language: English
• Published: Dordrecht SP MAIK Nauka/Interperiodica 01-09-2012; Springer Nature B.V
• Subjects: Analysis; Asymptotic properties; Boundary conditions; Boundary value problems; Density; Derivatives; Difference and Functional Equations; Differential equations; Diffusion; Dirichlet problem; Helmholtz equations; Integral equations; Mathematical analysis; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations; Planes; Studies; Neumann Problem; Dirichlet Condition; Singular Integral Equation; Compact Operator; Diffusion Equation
• ISSN: 0012-2661; 1608-3083
• DOI: 10.1134/S0012266112090017

We consider a boundary value problem for the stationary diffusion equation outside cuts on the plane. The Dirichlet condition is posed on one side of each cut, and an oblique derivative condition is posed on the other side. We prove existence and uniqueness theorems for the solution of the boundary value problem. We obtain an integral representation of a solution in the form of potentials. The densities of these potentials are found from a system of Fredholm integral equations of the second kind, which is uniquely solvable. We obtain closed asymptotic formulas for the gradient of the solution of the boundary value problem at the endpoints of the cuts.