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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.
35J45 Systems of elliptic equations, general (MSC2000)
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
58A14 Hodge theory in global analysis
47F05 General theory of partial differential operators
47L50 Dual spaces of operator algebras
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