Abstract:
In the work we study differential operators on arbitrary geometric graphs without loops. We extend
the known results for differential operators on an interval to the differential operators on the graphs. In order to define properly the maximal operator on a given graph, we need to choose a set of boundary vertices. The non-boundary vertices are called interior vertices. We stress that the maximal operator on a graph is determined not only by the given differential expressions on the edges, but also by the Kirchhoff conditions at the interior vertices of the graph. For the introduced maximal operator we prove an analogue of the Lagrange formula. We provide an algorithm for constructing adjoint boundary forms for an arbitrary set of boundary conditions. In the conclusion of the paper, we present a complete description of all self-adjoint restrictions of the maximal operator.
Keywords:
Directed graph, Kirchhoff conditions, self-adjoint restriction of an operator, maximal operator.
Citation:
L. K. Zhapsarbayeva, B. E. Kanguzhin, M. N. Konyrkulzhayeva, “Self-adjoint restrictions of maximal operator on graph”, Ufa Math. J., 9:4 (2017), 35–43
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\by L.~K.~Zhapsarbayeva, B.~E.~Kanguzhin, M.~N.~Konyrkulzhayeva
\paper Self-adjoint restrictions of maximal operator on graph
\jour Ufa Math. J.
\yr 2017
\vol 9
\issue 4
\pages 35--43
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\crossref{https://doi.org/10.13108/2017-9-4-35}
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Linking options:
https://www.mathnet.ru/eng/ufa412
https://doi.org/10.13108/2017-9-4-35
https://www.mathnet.ru/eng/ufa/v9/i4/p36
This publication is cited in the following 4 articles:
Baltabek KANGUZHİN, Karlygash DOSMAGULOVA, “Well-posed problems for the Laplace-Beltrami operator on a punctured two-dimensional sphere”, Advances in the Theory of Nonlinear Analysis and its Application, 7:2 (2023), 428
N. K. Shazhdekeeva, E. S. Zhakatai, “Postanovka kraevoi zadachi na grafe”, Mezhdunar. nauch.-issled. zhurn., 2020, no. 2(92), 40–43
Yerkebulan Zhakatay, Nurgul Shazhdekeeva, Adilet Raissov, “GRAFTAҒY ShETTІK ESEPTІ ZhҮIE ҮShІN ShETTІK ESEPKE TҮRLENDІRU / TRANSFORMATION OF A BOUNDARY VALUE PROBLEM ON A GRAPH INTO A BOUNDARY VALUE PROBLEM FOR A SYSTEM”, Vestnik, 2020
B. Kanguzhin, L. Zhapsarbaeva, Zh. Madibaiuly, “Lagrange formula for differential operators and self-adjoint restrictions of the maximal operator on a tree”, Eurasian Math. J., 10:1 (2019), 16–29
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