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Analysis of differential operators associated with boundary value problems. (English. Russian original) Zbl 1214.47038
J. Math. Sci., New York 159, No. 1, 4-46 (2009); translation from Probl. Mat. Anal. 40, 7-48 (2009).
In general, boundary value problems (BVP) for differential equations in vector bundles are studied by means of reducing the BVP to equations of traces of the solution on the boundary. The paper considers BVP for differential operators with transmission boundary conditions and impedance boundary conditions in a domain of the \(n\)-dimensional Euclidean space obtained by eliminating from \(\mathbb R^n\) the boundary \(\mathfrak S\). On both sides of the boundary \(\mathfrak S\), without the use of limit passages, operators of trace computation are constructed. Traces on the boundary, the Hilbert space of traces and the projection operators are defined first. Based on the projection operators and on the operators that define the transmission and impedance type boundary conditions, operators on the trace space are constructed. The author proves the invertibility of these operators, which further implies the unique solvability of the corresponding BVP. For the resolvents, Krein-type representations are obtained. Several simple examples of the application of the above results are shown. Further, after presenting some results on traces and von Neumann expansions, the author shows that, for a symmetric differential operator, the constructed resolvents of BVP correspond to closed von Neumann extensions of this operator.
MSC:
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
35P05 General topics in linear spectral theory for PDEs
47B25 Linear symmetric and selfadjoint operators (unbounded)
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