Boundary Integral Equations of the First Kind for the Heat Equation

Boundary element methods are being applied with increasing frequency to time dependent problems, especially to boundary value problems for parabolic differential equations. Here we shall consider the heat equation as the prototype of such equations. Various types of integral equations arise when solving boundary value problems for the heat equation. An important one is the single layer heat potential operator equation, i.e., the Volterra integral equation of the first kind with the fundamental solution as kernel. This equation is not well understood. The fundamental questions of existence and uniqueness of solutions and continuous dependence of the solution on the data have thus far not been answered. Such an investigation is basic. It must precede any rigorous analysis of the convergence of numerical methods for the equation. In this paper we shall set out the proper mathematical framework and establish the well-posedness of the single layer heat potential operator equation.