Boundary Integral Operators on Lipschitz Domains: Elementary Results

The simple and double layer potentials for second order linear strongly elliptic differential operators on Lipschitz domains are studied and it is shown that in a certain range of Sobolev spaces, results on continuity and regularity can be obtained without using either Calderón’s theorem on the L₂ -continuity of the Cauchy integral on Lipschitz curves J. L. Journé, “Calderón-Zygmuno operators, pseudo-differential operators and the Cauchy integral of Calderón, ” in Lecture Notes in Math. 994, Springer-Verlag, Berlin, 1983 or Dahlberg’s estimates of harmonic measures “On the Poisson integral for Lipschitz and C¹ domains, ” Studio Math. , 66 (1979), pp. 7–24. The operator of the simple layer potential and of the normal derivative of the double layer potential are shown to be strongly elliptic in the sense that they satisfy Cårding inequalities in the respective energy norms. As an application, error estimates for Galerkin approximation schemes for integral equations of the first kind are derived.