On a Solvability Condition for Systems with an Injective Symbol in Terms of Iterations of Double Layer Potentials

Abstract

We prove existence of the H ^p (D)-limit for iterations of the double layer potentials constructed from a Hodge parametrix on a smooth compact manifold X (here D is an open connected subset in X). The limit is the orthogonal projection of the Sobolev space H ^p (D) onto the closed subspace of H ^p (D)-solutions of some elliptic operator P of order p≥1. Using this result, we obtain a formula for Sobolev solutions to the equation Pu=f in D if such exist. Solutions are given in the form of series whose summands are iterations of the double layer potentials. We also construct a similar expansion for the Neumann P-problem in D.