Mixed problem for the Laplace equation outside cuts in a plane with setting Dirichlet and skew derivative conditions on different sides of the cuts

Krutitskii, P.; Sgibnev, A.

doi:10.1090/S0033-569X-06-00987-0

Authors: P. A. Krutitskii and A. I. Sgibnev
Journal: Quart. Appl. Math. 64 (2006), 105-136
MSC (2000): Primary 35J05, 45E05
DOI: https://doi.org/10.1090/S0033-569X-06-00987-0
Published electronically: January 24, 2006
MathSciNet review: 2211380
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Abstract: The mixed problem for the Laplace equation outside cuts in a plane is considered. The Dirichlet condition is posed on one side of each cut and the skew derivative condition is posed on the other side. This problem generalizes the mixed Dirichlet-Neumann problem. Integral representation for a solution of the boundary value problem is obtained in the form of potentials. The densities in the potentials satisfy the uniquely solvable Fredholm integral equation of the second kind and index zero. Uniqueness and existence theorems for a solution of the boundary value problem are proved. Singularities of the gradient of the solution of the boundary value problem at the tips of the cuts are studied. Asymptotic formulas for singularities are obtained. The effect of the disappearance of singularities is discussed.

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