When does a double-layer potential equal to a single-layer one?

2021/12/24 by Ramm, Alexander G.
#31A10 #35C05 #35J05 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2112.13095

Abstract

Let D be a bounded domain in ℝ³ with a closed, smooth, connected boundary S, N be the outer unit normal to S, k>0 be a constant, u_N^± are the limiting values of the normal derivative of u on S from D, respectively D':=ℝ³∖ D; g(x,y)=\frace^ik|x-y|4π|x-y|, w:=w(x,μ):=∫_S g_N(x,s)μ(s)ds be the double-layer potential, u:=u(x,σ):=∫_S g(x,s)σ(s)ds be the single-layer potential. In this paper it is proved that for every w there is a unique u, such that w=u in D and vice versa. Necessary and sufficient conditions are given for the existence of u and the relation w=u in D', given w in D', and for the existence of w and the relation w=u in D', given u in D'.

Related

• An inverse problem for the double layer potential
• A Singular Integral approach to a Two Phase Free Boundary Problem
• Eigenvalues and Eigenfunctions of Double Layer Potentials
• The adjoint double layer potential on smooth surfaces in ℝ³ and the Neumann problem
• A limiting case of a theorem of C. Miranda for layer potentials in Schauder spaces