The method of layer potentials for non-smooth domains with arbitrary topology. (English) Zbl 0915.35032

This paper is devoted to the study of boundary value problems for the Laplace operator in Lipschitz domains with arbitrary topology via boundary layers. First, two integral identities of Rellich type are employed for scalar-valued harmonic functions and for harmonic vector fields (§3). These and certain Fredholm operator properties (established in §4) are used to investigate invertibility properties of boundary integral operators, single layer and (singular) double layer potential operators (introduced in §2 and needed in the sequel). Those results are utilized in the last section (§5) in order to discuss solvability as well as regularity representation of solutions to three boundary value problems in bounded Lipschitz domains with arbitrary topology, namely, for the Dirichlet and Neumann problems for the Laplace operator and for a problem occurring in irrotational incompressible fluid flow (the Kelvin problem).


35J25 Boundary value problems for second-order elliptic equations
31B20 Boundary value and inverse problems for harmonic functions in higher dimensions
35Q35 PDEs in connection with fluid mechanics
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