, Volume 48, Issue 9, pp 1197-1211
Date: 25 Nov 2012

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

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Abstract

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.

Original Russian Text © P.A. Krutitskii, K.V. Prozorov, 2012, published in Differentsial’nye Uravneniya, 2012, Vol. 48, No. 9, pp. 1219–1233.