The double layer potential method for a boundary transmission problem for the Laplace operator in an infinite wedge

Abstract

Communicated by E. MeisterThis paper is concerned with the solution of a boundary transmission problem in an infinite wedge. We treat this problem by a boundary integral method using Green's contact function for two half-spaces. The integral operators are studied via a harmonic analysis approach which goes back to a paper of Fabes et al. We improve their results studying the Fourier symbol of the associated integral operators on the half-plane. This leads to invertibility criteria for the boundary integral operators.

‐solution of the Neumann, Robin and transmission problem for the scalar Oseen equation

“…It means that we look for a solution such that the nontangential maximal function of and ∇ are in ( Ω) and boundary conditions are fulfilled in the sense of nontangential limits. The goal of the paper is to prove results similar to the well-known results corresponding to the Laplace equation ( [4,5,[8][9][10][11][18][19][20][21]23,30,31,33,37,38,40,41,46,48]). We find necessary and sufficient conditions for the existence of an -solution of the Neumann and Robin problem for bounded and unbounded domains with compact Lipschitz boundary.…”

On the Boundedness of Some Integral Operators With Fixed Singularities Arising in Elasticity

“…In both cases (10), (12) we have the same integral operator with the kernel (11). In order to know the structure of K(x,y) we have to look at V(x,y).…”

“…In the plane case the BIEs are locally considered in the neighbourhood of an interface corner of Mellin convolution type and with the Mellin technique the question of Fredholm property and asymptotics can be decided [8]. In the spatial case D. Mirschinka [9,10] investigated the BIEs for bimetal heat conduction problems with perfect and non-perfect heat contact. The question of Fredholm property of the matrix boundary integral operator is connected with the invertibility of local operators acting in spaces of functions defined on the tangential half-planes to S at points of the interface edge dSoo-The local operators have fixed singularities along the boundary of the half-plane and can be written as convolution operators.…”