On the spectral properties of a non-coercive mixed problem associated with \(\overline\partial \)-operator. (English) Zbl 1476.35105
Summary: 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.
MSC:
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35P10 | Completeness of eigenfunctions and eigenfunction expansions in context of PDEs |
| 34B24 | Sturm-Liouville theory |
| 47N20 | Applications of operator theory to differential and integral equations |
Keywords:
Sturm-Liouville problem; non-coercive problems; the multidimensional Cauchy-Riemann operator; root functionsReferences:
| [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 · Zbl 0093.10401 · doi:10.1002/cpa.3160120405 |
| [2] | J. J. Kohn, L. Nirenberg, “Non-coercive boundary value problems”, Comm. Pure Appl. Math., 18 (1965), 443-492 · Zbl 0125.33302 · doi:10.1002/cpa.3160180305 |
| [3] | J. J. Kohn, “Subellipticity of the \(\overline\partial \)-Neumann problem on pseudoconvex domains: sufficient conditions”, Acta Math., 142:1-2 (1979), 79-122 · Zbl 0395.35069 · doi:10.1007/BF02395058 |
| [4] | L. Hörmander, “Pseudo-differential operators and non-elliptic boundary value problems”, Ann. Math., 83:1 (1966), 129-209 · Zbl 0132.07402 · doi:10.2307/1970473 |
| [5] | I. Ts. Gokhberg, M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Spaces, AMS, Providence, R.I., 1969 · Zbl 0181.13503 |
| [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) · Zbl 0045.39402 |
| [7] | S. Agmon, “On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems”, Comm. Pure Appl. Math., 15 (1962), 119-147 · Zbl 0109.32701 · doi:10.1002/cpa.3160150203 |
| [8] | F. E. Browder, “On the spectral theory of strongly elliptic differential operators”, Proc. Nat. Acad. Sci. USA, 45 (1959), 1423-1431 · Zbl 0087.30403 · doi:10.1073/pnas.45.9.1423 |
| [9] | M. S. Agranovich, “Spectral Problems in Lipschitz Domains”, Modern Mathematics, Fundamental Trends, 39, 2011, 11-35 (Russian) |
| [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 · Zbl 1059.35519 |
| [11] | N. Tarkhanov, “On the root functions of general elliptic boundary value problems”, Compl. Anal. Oper. Theory, 1 (2006), 115-141 · Zbl 1211.35209 · doi:10.1007/s11785-006-0003-9 |
| [12] | M. S. Agranovich, “Mixed problems in a Lipschitz domain for strongly elliptic second order systems”, Funct. Anal. Appl., 45:2 (2011), 81-98 · Zbl 1271.35024 · doi:10.1007/s10688-011-0011-z |
| [13] | O. A. Ladyzhenskaya, N. N. Uraltseva, Linear and Quasilinear Equations of Elliptic Type, Nauka, Moscow, 1973 (in Russian) · Zbl 0269.35029 |
| [14] | J. L. Lions, E. Magenes, Non-Homogeneous Boundary Value Problems und Applications, v. 1, Springer-Verlag, Berlin-Heidelberg-New York, 1972 · Zbl 0227.35001 |
| [15] | M. Schechter, “Negative norms and boundary problems”, Ann. Math., 72:3 (1960), 581-593 · Zbl 0097.08401 · doi:10.2307/1970230 |
| [16] | N. Tarkhanov, The Cauchy Problem for Solutions of Elliptic Equations, Akademie-Verlag, Berlin, 1995 · Zbl 0831.35001 |
| [17] | M. S. Agranovich, “Elliptic operators on closed manifold”, Current Problems of Mathematics, Fundamental Directions, 63, VINITI, 1990, 5-129 (Russian) · Zbl 0739.58061 |
| [18] | H. Triebel, Interpolation Theory, Function spaces, Differential operators, VEB Wiss. Verlag, Berlin, 1978 · Zbl 0387.46033 |
| [19] | A. N. Tikhonov, A. A. Samarskii, Equations of Mathematical Physics, Nauka, Moscow, 1972 (in Russian) |
| [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 · Zbl 0782.30003 |
| [21] | A. A. Shlapunov, N. Tarkhanov, “Mixed problems with a parameter”, Russ. J. Math. Phys., 12:1 (2005), 97-124 · Zbl 1190.58020 |
| [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 · Zbl 0955.35017 |
| [23] | A. N. Kolmogorov, S. V. Fomin, Elements of Function’s Theory and Functional Analysis, Nauka, Moscow, 1989 (in Russian) · Zbl 0672.46001 |
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.
