On the Solvability of a Boundary Value Problem for the Laplace Equation on a Screen with a Boundary Condition for a Directional Derivativeстатья
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Дата последнего поиска статьи во внешних источниках: 27 ноября 2016 г.
Аннотация:We consider a three-dimensional boundary value problem for the Laplace equation
on a thin plane screen with boundary conditions for the “directional derivative”: boundary
conditions for the derivative of the unknown function in the directions of vector fields defined on
the screen surface are posed on each side of the screen. We study the case in which the direction
of these vector fields is close to the direction of the normal to the screen surface. This problem
can be reduced to a system of two boundary integral equations with singular and hypersingular
integrals treated in the sense of the Hadamard finite value. The resulting integral equations
are characterized by the presence of integral-free terms that contain the surface gradient of one
of the unknown functions. We prove the unique solvability of this system of integral equations
and the existence of a solution of the considered boundary value problem and its uniqueness
under certain assumptions.