Abstract:
Differential equations of the following forms are considered:
y′+Ay=0and−y″+A2y=0,
where A is a positive selfadjoint operator in a Hilbert space H. The question of whether the solutions of such equations have boundary values at the end points of the interval (a,b) on which they are considered is investigated, as well as the problem of recovering a solution from its boundary values. A characterization of the boundary values is given in terms of the behavior of the solution near the end points a and b. A number of examples are cited in which A is realized as a differential operator in various function spaces. When applied to these concrete situations, the abstract theorems yield the existence and characteristics of the boundary values for certain classes of elliptic and parabolic equations; in particular, the well-known results of F. Riesz, Köthe and Komatsu are obtained and sharpened in this way. The approach is based on the spectral theory of selfadjoint operators.
Bibliography: 18 titles.
Citation:
V. I. Gorbachuk, M. L. Gorbachuk, “Boundary values of solutions of some classes of differential equations”, Math. USSR-Sb., 31:1 (1977), 109–133
\Bibitem{GorGor77}
\by V.~I.~Gorbachuk, M.~L.~Gorbachuk
\paper Boundary values of solutions of some classes of differential equations
\jour Math. USSR-Sb.
\yr 1977
\vol 31
\issue 1
\pages 109--133
\mathnet{http://mi.mathnet.ru/eng/sm2641}
\crossref{https://doi.org/10.1070/SM1977v031n01ABEH002293}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=486970}
\zmath{https://zbmath.org/?q=an:0356.34066|0392.34040}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=A1977FV43600007}
Linking options:
https://www.mathnet.ru/eng/sm2641
https://doi.org/10.1070/SM1977v031n01ABEH002293
https://www.mathnet.ru/eng/sm/v144/i1/p124
This publication is cited in the following 16 articles:
Horbachuk V.M., Horbachuk M.L., “Spaces of Smooth and Generalized Vectors of the Generator of An Analytic Semigroup and Their Applications”, Ukr. Math. J., 69:4 (2017), 561–597
Gorbachuk M.L., Gorbachuk V.I., “On Behavior of Weak Solutions of Operator Differential Equations on (0, Infinity)”, Modern Analysis and Applications: Mark Krein Centenary Conference, Vol 2, Operator Theory Advances and Applications, 191, eds. Adamyan V., Berezansky Y., Gohberg I., Gorbachuk M., Gorbachuk V., Kochubei A., Langer H., Popov G., Birkhauser Verlag Ag, 2009, 115–126
Litovchenko V.A., “The Cauchy Problem for a Class of Evolution Systems of the Parabolic Type with Unbounded Coefficients”, Differ. Equ., 44:6 (2008), 835–854
M. L. Gorbachuk, V. I. Gorbachuk, Algebraic and Geometric Methods in Mathematical Physics, 1996, 425
V. I. Gorbachuk, A. V. Knyazyuk, “Boundary values of solutions of operator-differential equations”, Russian Math. Surveys, 44:3 (1989), 67–111
Bondarenko N., “Self-Conjugate Expansions of the Minimal Operator Generated by the Degenerating Differential-Operator Sturm-Liouville Equation”, no. 8, 1987, 3–6
Gorbachuk M., Pivtorak N., “Solutions of Evolution Parabolic Equations with Degeneration”, Differ. Equ., 21:8 (1985), 892–897
V. V. Gorodetskii, M. L. Gorbachuk, “Polynomial approximation of solutions of operator-differential equations in a Hilbert space”, Ukr Math J, 36:4 (1985), 409
Knyazyuk A., “Boundary-Values of Solutions of Differential-Equations in the Banach-Space”, no. 9, 1984, 12–13
Fedorova L., “Boundary-Values of Solutions for Inhomogeneous Differentially-Operator Equations”, no. 7, 1983, 21–24
Suleimanov N., “One of the Refinements of Wiman-Valiron-Type Estimates for Solutions of Evolutional Equations”, 265, no. 3, 1982, 545–546
N. M. Suleimanov, “Theorems of Wiman–Valiron type for solutions of parabolic equations”, Math. USSR-Sb., 43:1 (1982), 63–84
Gorbachuk V., Gorbachuk M., “Trigonometric Series and Generalized Periodic-Functions”, 257, no. 4, 1981, 799–804
Gorbachuk M., Dudnikov P., “On Initial Data of the Cauchy-Problem for Parabolic Equations for Which the Solution Is Infinitely Differentiable”, no. 4, 1981, 9–11
Suleimanov N., “On the Exactness of Wiman-Valiron-Type Estimates for the Solutions of Evolutionary Equations”, 253, no. 3, 1980, 541–544
Kashpirovsky A., “Analytic Representation of Generalized-Functions of S'-Type”, no. 4, 1980, 12–14
Citation in format AMSBIB
Contact us: