Reduction of Boundary-Value Problems for the Laplace Equation to Boundary Equations of Calderón—Seeley Type

Abstract

It will be shown in this chapter how to replace the Laplace equation in a domain by equivalent relations connecting the values of the desired solution $u|_\Gamma$ and its normal derivative $\frac{{\partial u}} {{\partial n}}\left| {_\Gamma } \right.$ on the boundary of the domain. These relations, together with the boundary conditions defining the boundary-value problem, form a system of equations for functions defined only on the boundary equivalent to the original boundary-value problem for the Laplace equation in the domain.