Abstract
For problems of transmission for higher-order equations with inhomogeneous conjugation conditions, we construct and investigate the corresponding operators in Hilbert spaces. We prove the self-adjointness of these operators for a spectral problem with a parameter in conjugation conditions.
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REFERENCES
V. I. Il'in, “On the solvability of the Dirichlet and Neumann problems for a linear elliptic operator with discontinuous coefficients,” Dokl. Akad. Nauk SSSR, 137, No. 1, 28–31 (1961).
M. Schechter, “A generalization of the problem of transmission,” Ann. Soc. Norm. Supér. Pisa, 14, 207–236 (1960).
Z. G. Sheftel', “Energy inequalities and general boundary-value problems for elliptic equations with discontinuous coefficients,” Sib. Mat. Zh., 6, No. 3, 636–668 (1965).
Ya. A. Roitberg, “Theorem on homeomorphisms realized by elliptic operators and local increase in the smoothness of generalized solutions,” Ukr. Mat. Zh., 17, No. 5, 122–129 (1965).
B. Ya. Roitberg, “Problems of transmission in domains with nonsmooth boundaries,” Dopov. Nats. Akad. Nauk Ukr., No. 3, 15–20 (1996).
M. S. Agranovich and R. Mennicken, “Spectral problems for the Helmholtz equation with a spectral parameter in boundary conditions on a nonsmooth surface,” Mat. Sb., 190, No. 1, 29–68 (1999).
V. S. Deineka, I. V. Serhienko, and V. V. Skopets'kyi, “Eigenvalue problems with discontinuous eigenfunctions and their numerical solutions,” Ukr. Mat. Zh., 51, No. 10, 1317–1323 (1999).
O. N. Komarenko and V. A. Trotsenko, “Variational method for the solution of problems of transmission with the principal conjugation condition,” Ukr. Mat. Zh., 51, No. 6, 762–775 (1999).
J. Odnoff, “Operator generated by differential problems with eigenvalue parameter in equation and boundary condition,” Medd. Lunds. Univ. Math. Semin., 14, 35–69 (1959).
A. N. Komarenko, I. A. Lukovskii, and S. F. Feshchenko, “On the eigenvalue problem with a parameter in boundary conditions,” Ukr. Mat. Zh., 17, No. 6, 740–748 (1965).
J. Ercolano and M. Schechter, “Spectral theory for operators generated by elliptic boundary problems,” Commun. Pure Appl. Math, 18, No. 1, 2, 83–105 (1965).
V. V. Barkovskii and Ya. A. Roitberg, “On minimum and maximum operators corresponding to the general elliptic problem with inhomogeneous boundary conditions,” Ukr. Mat. Zh., 18, No. 2, 91–97 (1966).
V. V. Barkovskii, “Self-adjointness of operators generated by a general elliptic expression and inhomogeneous boundary conditions imposed on a part of the boundary of the bounded domain,” Ukr. Mat. Zh., 22, No. 4, 527–531 (1970).
Yu. M. Berezanskii and Ya. A. Roitberg, “Theorem of homeomorphisms and Green function for general elliptic boundary-value problems,” Ukr. Mat. Zh., 19, No. 5, 3–32 (1967).
L. P. Nizhnik and L. A. Taraborkin, “On self-adjoint operators generated by inhomogeneous elliptic problems with discontinuous boundary conditions and conjugation conditions,” Ukr. Mat. Zh., 43, No. 3, 374–381 (1991).
L. P. Nizhnik and L. A. Taraborkin, “Evolution problem for harmonic functions in a domain with thin inclusion,” Dopov. Nats. Akad. Nauk Ukr., No. 1, 13–16 (1994).
Yu. M. Berezanskii, Expansion in Eigenfunctions of Self-Adjoint Operators [in Russian], Naukova Dumka, Kiev (1965).
J.-L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications, Dunod, Paris (1968).
I. Ya. Roitberg and Ya. A. Roitberg, “Green formulas for general elliptic boundary-value problems for systems of Douglis-Nirenberg structure,” Dokl. Ros. Akad. Nauk, 359, No. 6, 739–743 (1998).
Ya. A. Roitberg, Boundary Value Problems in the Spaces of Distributions, Kluwer, Dordrecht (1999).
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Komarenko, O.N. Self-Adjoint Operators Generated by Problems of Transmission with Inhomogeneous Conjugation Conditions. Ukrainian Mathematical Journal 53, 1956–1975 (2001). https://doi.org/10.1023/A:1015434821913
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DOI: https://doi.org/10.1023/A:1015434821913