Geodesic
On the spectral properties of a~non-coercive mixed problem associated with $\overline\partial$-operator
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 6 (2013) no. 2, pp. 247-261.

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We consider a non-coercive Sturm–Liouville boundary value problem in a bounded domain $D$ of the complex space $\mathbb C^n$ for the perturbed Laplace operator. More precisely, the boundary conditions are of Robin type on $\partial D$ while the first order term of the boundary operator is the complex normal derivative. We prove that the problem is Fredholm one in proper spaces for which an Embedding Theorem is obtained; the theorem gives a correlation with the Sobolev–Slobodetskii spaces. Then, applying the method of weak perturbations of compact self-adjoint operators, we show the completeness of the root functions related to the boundary value problem in the Lebesgue space. For the ball, we present the corresponding eigenvectors as the product of the Bessel functions and the spherical harmonics.
Keywords: non-coercive problems, the multidimensional Cauchy–Riemann operator, root functions.
Mots-clés : Sturm–Liouville problem 
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Alexander N. Polkovnikov; Aleksander A. Shlapunov. On the spectral properties of a~non-coercive mixed problem associated with $\overline\partial$-operator. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 6 (2013) no. 2, pp. 247-261. http://geodesic.mathdoc.fr/item/JSFU_2013_6_2_a11/

[1] S. Agmon, A. Douglis, L. Nirenberg, “Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Part 1”, Comm. Pure Appl. Math., 12 (1959), 623–727 | DOI | MR | Zbl

[2] J. J. Kohn, L. Nirenberg, “Non-coercive boundary value problems”, Comm. Pure Appl. Math., 18 (1965), 443–492 | DOI | MR | Zbl

[3] J. J. Kohn, “Subellipticity of the $\overline\partial$-Neumann problem on pseudoconvex domains: sufficient conditions”, Acta Math., 142:1–2 (1979), 79–122 | DOI | MR | Zbl

[4] L. Hörmander, “Pseudo-differential operators and non-elliptic boundary value problems”, Ann. Math., 83:1 (1966), 129-209 | DOI | MR

[5] I. Ts. Gokhberg, M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Spaces, AMS, Providence, R.I., 1969 | Zbl

[6] M. V. Keldysh, “On the characteristic values and characteristic functions of certain classes of non-selfadjoint equations”, Dokl. AN SSSR, 77 (1951), 11–14 (in Russian) | MR | Zbl

[7] S. Agmon, “On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems”, Comm. Pure Appl. Math., 15 (1962), 119–147 | DOI | MR | Zbl

[8] F. E. Browder, “On the spectral theory of strongly elliptic differential operators”, Proc. Nat. Acad. Sci. USA, 45 (1959), 1423–1431 | DOI | MR | Zbl

[9] M. S. Agranovich, “Spectral Problems in Lipschitz Domains”, Modern Mathematics, Fundamental Trends, 39, 2011, 11–35 (Russian) | MR

[10] V. A. Kondrat'ev, “Completeness of the systems of root functions of elliptic operators in Banach spaces”, Russ. J. Math. Phys., 6:10 (1999), 194–201 | MR

[11] N. Tarkhanov, “On the root functions of general elliptic boundary value problems”, Compl. Anal. Oper. Theory, 1 (2006), 115–141 | DOI | MR

[12] M. S. Agranovich, “Mixed problems in a Lipschitz domain for strongly elliptic second order systems”, Funct. Anal. Appl., 45:2 (2011), 81–98 | DOI | MR

[13] O. A. Ladyzhenskaya, N. N. Uraltseva, Linear and Quasilinear Equations of Elliptic Type, Nauka, Moscow, 1973 (in Russian) | MR

[14] J. L. Lions, E. Magenes, Non-Homogeneous Boundary Value Problems und Applications, v. 1, Springer-Verlag, Berlin–Heidelberg–New York, 1972

[15] M. Schechter, “Negative norms and boundary problems”, Ann. Math., 72:3 (1960), 581–593 | DOI | MR | Zbl

[16] N. Tarkhanov, The Cauchy Problem for Solutions of Elliptic Equations, Akademie-Verlag, Berlin, 1995 | MR | Zbl

[17] M. S. Agranovich, “Elliptic operators on closed manifold”, Current Problems of Mathematics, Fundamental Directions, 63, VINITI, 1990, 5–129 (Russian) | MR | Zbl

[18] H. Triebel, Interpolation Theory, Function spaces, Differential operators, VEB Wiss. Verlag, Berlin, 1978 | MR

[19] A. N. Tikhonov, A. A. Samarskii, Equations of Mathematical Physics, Nauka, Moscow, 1972 (in Russian) | MR

[20] L. A. Aizenberg, A. M. Kytmanov, “On the possibility of holomorphic continuation to a domain of functions given on a part of its boundary”, Mat. sb., 182:4 (1991), 490–507 | MR | Zbl

[21] A. A. Shlapunov, N. Tarkhanov, “Mixed problems with a parameter”, Russ. J. Math. Phys., 12:1 (2005), 97–124 | MR

[22] A. A. Shlapunov, “Spectral decomposition of Green's integrals and existence of $W^{s,2}$-solutions of matrix factorizations of the Laplace operator in a ball”, Rend. Sem. Mat. Univ. Padova, 96 (1996), 237–256 | MR | Zbl

[23] A. N. Kolmogorov, S. V. Fomin, Elements of Function's Theory and Functional Analysis, Nauka, Moscow, 1989 (in Russian) | MR | Zbl