Coercivity of the Single Layer Heat Potential

Abstract

The single layer heat potential operator, K, arises in the solution of initial-boundary value problems for the heat equation using boundary integral methods. In this note we show that K maps a certain anisotropic Sobolev space isomorphically onto its dual, and, moreover, satisfies the coercivity inequality $ < K_{q,q} >\geq c\|q\|^2$. We thereby establish the well-posedness of the operator equation $K_q=f$ and provide a basis for the analysis of the discretizations.