BOUNDARY INTEGRAL EQUATIONS OF THE FIRST KIND FOR THE HEAT EQUATION D. N. Arnold and P. J. Noon Department of Mathematics, University of Maryland, College Park, MD 20742, U.S.A. INTRODUCTION 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 so- lution as kernel. This equation is not well understood. The fundamental questions of existence and uniqueness of solutions and continuous depen- dence 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 con- vergence 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. We begin by recalling how this equation arises. The direct method for deriving an integral equation formulation for transient heat conduction begins with a representation of the temperature at any point in the spatial domain Ω ⊂ R 3 and any positive time in terms of the temperature and flux on the boundary for all previous times and the initial temperature. Let u(x,t) denote the temperature at a point x in Ω, the spatial domain, and a time t ≥ 0, and assume that the thermal diffusivity is scaled to unity, so that u satisfies the heat equation ∂u ∂t (x,t) − Δ u(x,t)=0, x ∈ Ω,t> 0. Denote by K (x,y) the fundamental solution for the heat equation, K (x,t)=      exp(−|x| 2 /4t) (4πt) 3/2 ,x ∈ R 3 ,t> 0, 0, x ∈ R 3 ,t ≤ 0. (1)