Justification of Fabrikant's method for solving mixed problems of potential theory (Q1178546)

The present paper provides a rigorous justification of a new boundary integral method introduced recently by the second author. The key to the analysis is the redefinition of the so-called \(L(k)\)-operator (Poisson type operator) involved in the integral representation. To demonstrate Fabrikant's method, the authors apply this method to the solution of the Laplace equation \(\Delta V=0\) in \(R^ 3_ +=\{(x,y,z):z>0\}\) under the mixed boundary conditions \(V=v(r,\varphi)\) if \(r<a\), \(V_ z=0\) if \(r>a\) \((z=0)\), and \(V(\infty)=0\), where \((r,\varphi)\) are the polar coordinates in the plane \(P=\{(x,y,z):z=0\}\), \(a=\text{const}>0\) denotes a given positive constant and \(v\) is a given function in the disk \(D=\{(x,y,0):r<a\}\).