Geodesic
On the Spectrum of Well-Defined Restrictions and Extensions for the Laplace Operator
Matematičeskie zametki, Tome 95 (2014) no. 4, pp. 507-516.

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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. But 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 $\widehat{L}$ 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 $\widehat{L}$ 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.
Keywords: Laplace operator, maximal (minimal) operator, Volterra operator, Volterra well-defined restrictions and extensions of operators, Hilbert space, elliptic operator, Poisson operator. 
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B. N. Biyarov. On the Spectrum of Well-Defined Restrictions and Extensions for the Laplace Operator. Matematičeskie zametki, Tome 95 (2014) no. 4, pp. 507-516. http://geodesic.mathdoc.fr/item/MZM_2014_95_4_a2/

[1] M. I. Vishik, “Ob obschikh kraevykh zadachakh dlya ellipticheskikh differentsialnykh uravnenii”, Tr. MMO, 1, GITTL, M.–L., 1952, 187–246 | MR | Zbl

[2] A. V. Bitsadze, A. A. Samarskii, “O nekotorykh prosteishikh obobscheniyakh lineinykh ellipticheskikh kraevykh zadachakh”, Dokl. AN SSSR, 185:4 (1969), 739–740 | MR | Zbl

[3] A. A. Dezin, Partial Differential Equations. An Introduction to a General Theory of Linear Boundary Value Problems, Springer Ser. Soviet Math., Springer-Verlag, Berlin, 1987 | MR | Zbl

[4] B. K. Kokebaev, M. Otelbaev, A. N. Shynybekov, “O rasshireniyakh i suzheniyakh operatorov v banakhovom prostranstve”, UMN, 37:4 (1982), 116

[5] L. Khermander, K teorii obschikh differentsialnykh operatorov v chastnykh proizvodnykh, IL, M., 1959

[6] B. N. Biyarov, Spektralnye svoistva korrektnykh suzhenii i rasshirenii, (LAP) LAMBERT Acad. Publ., Saarbrucken, 2012

[7] M. Engliš, “Analytic continuation of weighted Bergman kernels”, J. Math. Pures Appl. (9), 94:6 (2010), 622–650 | MR | Zbl

[8] G. Grubb, “Singular Green operators and their spectral asymptotics”, Duke Math. J., 51:3 (1984), 477–528 | DOI | MR | Zbl

[9] H. Triebel, “Approximation numbers in function spaces and the distribution of eigenvalues of some fractal elliptic operators”, J. Approx. Theory, 129:1 (2004), 1–27 | DOI | MR | Zbl

[10] A. Jonsson, H. Wallin, Function Spaces on Subsets of $\mathbb{R}^n$, Math. Rep., 2, no. 1, Harwood Acad. Publ., London, 1984 | MR | Zbl

[11] K. J. Falconer, The Geometry of Fractal Sets, Cambridge Tracts in Math., 85, Cambridge Univ. Press, Cambridge, 1985 | MR | Zbl

[12] N. Danford, Dzh. Shvarts, Lineinye operatory. T. 2. Spektralnaya teoriya. Samosopryazhennye operatory v gilbertovom prostranstve, IL, M., 1966 | MR | Zbl

[13] I. Ts. Gokhberg, M. G. Krein, Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov v gilbertovom prostranstve, Nauka, M., 1965 | MR | Zbl