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A generalization of the Neumann problem for the Helmholtz equation outside cuts on the plane. (English. Russian original) Zbl 1143.35019
Differ. Equ. 41, No. 9, 1213-1224 (2005); translation from Differ. Uravn. 41, No. 9, 1155-1165 (2005).
The authors deal with a problem that generalizes the Neumann problem for the Helmholtz equation outside cuts on the plane with boundary conditions occurring on the contact surface in some physical problems. The uniqueness of the solution of the problem under consideration is proved based on the energy identities. Using a single layer potential and angular potential, the authors reduce the problem to a singular integral equation of the first kind with additional conditions, which, in turn is reduced to a uniquely solvable Fredholm integral equation of the second kind by some transformations. Moreover, the authors obtain formulas for the singularities of the gradient of the solution at the endpoints of the cuts in closed form.

MSC:
35J25 Boundary value problems for second-order elliptic equations
35C05 Solutions to PDEs in closed form
45B05 Fredholm integral equations
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