×

On the spectrum of well-defined restrictions and extensions for the Laplace operator. (English. Russian original) Zbl 1318.47059

Math. Notes 95, No. 4, 463-470 (2014); translation from Mat. Zametki 95, No. 4, 507-516 (2014).
Summary: The study of the spectral properties of operators generated by differential equations of hyperbolic or parabolic type with Cauchy initial data involve, as a rule, Volterra boundary-value problems that are well posed. However, Hadamard’s example shows that the Cauchy problem for the Laplace equation is ill-posed. At present, not a single Volterra well-defined restriction or extension for elliptic-type equations is known. Thus, the following question arises: Does there exist a Volterra well-defined restriction of a maximal operator \(\hat L\) or a Volterra well-defined extension of a minimal operator \(L_0\) generated by the Laplace operator? In the present paper, for a wide class of well-defined restrictions of the maximal operator \(\hat L\) and of well-defined extensions of the minimal operator \(L_0\) generated by the Laplace operator, we prove a theorem stating that they cannot be Volterra.

MSC:

47F05 General theory of partial differential operators
47B07 Linear operators defined by compactness properties
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M. I. Vishik, ”On general boundary problems for elliptic differential equations,” Trudy Moskov. Mat. Obshch. (GITTL, Moscow-Leningrad, 1952), Vol. 1, pp. 187–246 [Am. Math. Soc. Transl., II. Ser. 24, 107–172 (1963)].
[2] A. V. Bitsadze and A. A. Samarskii, ”On some simple generalizations of linear elliptic boundary problems,” Dokl. Akad. Nauk SSSR 185(4), 739–740 (1969) [Soviet Math. Dokl. 10, 398–400 (1969)]. · Zbl 0187.35501
[3] A. A. Dezin, Partial Differential Equations: An Introduction to a General Theory of Linear Boundary-Value Problems, in Springer Ser. Soviet Math. (Springer-Verlag, Berlin, 1987). · Zbl 0623.35005
[4] B. K. Kokebaev, M. Otelbaev, and A. N. Shynybekov, ”On the extensions and restrictions of operators in a Banach space,” Uspekhi Mat. Nauk 37(4), 116 (1982) · Zbl 0509.47054
[5] L. Hörmander, On the Theory of General Partial Differential Operators (Acta Math., 1955; Inostr. Lit., Moscow, 1959).
[6] B. N. Biyarov, Spectral Properties of Correct Restrictions and Extensions (Lambert Acad. Publ., Saarbrücken, 2012). · Zbl 0853.34030
[7] M. Engliš, ”Analytic continuation weighted Bergman kernels,” J. Math. Pures Appl. (9) 94(6), 622–650 (2010). · Zbl 1215.32005 · doi:10.1016/j.matpur.2010.08.004
[8] G. Grubb, ”Singular green operators and their spectral asymptotics,” Duke Math. J. 51(3), 477–528 (1984). · Zbl 0553.58034 · doi:10.1215/S0012-7094-84-05125-1
[9] H. Triebel, ”Approximation numbers in function spaces and the distribution eigenvalues of some fractal elliptic operators,” J. Approx. Theory 129(1), 1–27 (2004). · Zbl 1130.46305 · doi:10.1016/j.jat.2004.05.003
[10] A. Jonsson and H. Wallin, Function Spaces on Subsets of \(\mathbb{R}\)n, in Math. Rep. (Harwood Acad. Publ., London, 1984), Vol. 2 [in Russian]. · Zbl 0875.46003
[11] K. J. Falconer, Geometry of Fractal Sets, in Cambridge Tracts in Math. (Cambridge Univ. Press, Cambridge, 1985), Vol. 85 [in Russian].
[12] N. Dunford and J. T. Schwartz, Linear Operators, Vol. 2: Spectral Theory (Interscience Publ., New York, 1963; Mir, Moscow, 1966). · Zbl 0128.34803
[13] I. Ts. Gokhberg and M. G. Krein, Introduction to the Theory of Linear Non-Self-Adjoint Operators in Hilbert space (Nauka, Moscow, 1965) [in Russian].
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.