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The Neumann problem for the equation \(\Delta_u-k^2u=0\) in the exterior of non-closed Lipschitz surfaces. (English) Zbl 1286.35091

Summary: We study the Neumann problem for the equation \(\Delta_u-k^2u=0\) in the exterior of non-closed Lipschitz surfaces in \(\mathbb R^3\). Theorems on existence and uniqueness of a weak solution of the problem are proved. The integral representation for a solution is obtained in the form of a double layer potential. The density in the potential is defined as a solution of the operator (integral) equation, which is uniquely solvable.

MSC:

35J25 Boundary value problems for second-order elliptic equations
31A10 Integral representations, integral operators, integral equations methods in two dimensions
31A25 Boundary value and inverse problems for harmonic functions in two dimensions
35D30 Weak solutions to PDEs
35C15 Integral representations of solutions to PDEs
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References:

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