On a solvability condition for systems with an injective symbol in terms of iterations of double layer potentials.

*(Russian, English)*Zbl 1014.35022
Sib. Mat. Zh. 42, No. 4, 952-963 (2001); translation in Sib. Math. J. 42, No. 4, 801-810 (2001).

The author studies solvability conditions for the operator equation \(Pu=f\) in a compact manifold \(X\), where \(P\) is a linear differential operator with injective symbol. To solve the operator equation, the author uses the Hodge theory to construct the orthogonal projection acting from the Sobolev space \(H^p(D)\) onto a closed subspace of \(H^p(D)\)-solutions to the equation \(Pu =0\) in \(D\) where \(D\subset X\) is an open connected set, \(p\) denotes the order of the operator \(P\). A solution to the problem is given in the form of a series with terms presented by iterations of double layer potentials whereas a solvability condition for the above nonhomogeneous equation is equivalent to the convergence of this series together with orthogonality to \(\text{ker} P^*\). Moreover, the approach presented makes it possible to construct a similar resolution to the \(P\)-Neumann problem. The author gives applications of this approach to the Cauchy-Riemann system in \(\mathbb{C}^n\), \(n\geq 2\), the linear elasticity equations, and others.

Reviewer: V.Grebenev (Novosibirsk)