Abstract
We are concerned here with the spectral theory pertaining to an elliptic boundary value problem involving an indefinite weight function, or equivalently, the spectral theory for a pencil of the form A — λT acting in a Hilbert space L 2(Ω), where Ω ⊂ ℝn is a bounded region and n ≥ 2. Here A is a non-self adjoint operator and T is a multiplication operator in L 2(Ω) induced by a real-valued weight function which assumes both positive and negative values. Results are given concerning the completeness of the principal vectors of the pencil in certain function spaces as well as concerning the angular and asymptotic distribution of the eigenvalues. Furthermore, a new result is also derived pertaining to the asymptotic distribution of the eigenvalues.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
R.A Adams, Sobolev spaces, Academic, New York, 1975.
S. Agmon, On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems, Comm. Pure Appl. Math. 15 (1962), 119–147.
S. Agmon, Lectures on elliptic boundary value problems, Van Nostrand, Princeton, N.J., 1965.
S. Agmon, On kernels, eigenvalues, and eigenf unctions of operators related to elliptic problems, Comm. Pure Appl. Math. 18 (1965), 627–663.
S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623–727.
W. Allegretto and A.B. Mingarelli, Boundary problems of the second order with an indefinite weight-function, J. Reine Angew. Math. 398 (1989), 1–24.
R. Beals, Asymptotic behaviour of the Green’s function and spectral function of an elliptic operator, J. Funct. Anal. 5 (1970), 484–503.
R. Beals, Indefinite Sturm-Liouville problems and half-range completeness, J. Differential Equations 56 (1985), 391–407.
P. Binding and B. Najman, A variational principle in Krein spaces (preprint).
M.S. Birman and M.Z. Solomjak, Asymptotic behaviour of the spectrum of differential equations, J. Soviet Math. 12 (1974), 247–282.
M.S. Birman and M.Z. Solomjak, Asymptotics of the spectrum of variational problems on solutions of elliptic equations, Siberian Math. J. 20 (1979), 1–15.
J. Bognár, Indefinite inner product spaces, Springer, Berlin, 1974.
B. Ćurgus and B. Najman, A Krein space approach to elliptic eigenvalue problems with indefinite weights (preprint).
N. Dunford and J.T. Schwartz, Linear operators, part I, Wiley, New York, 1988.
M. Faierman, On the eigenvalues of nons elf adjoint problems involving indefinite weights, Math. Ann. 282 (1988), 369–377.
M. Faierman, Elliptic problems involving an indefinite weight, Trans. Amer. Math. Soc. 320 (1990), 253–279.
M. Faierman, Non-self adjoint elliptic problems involving an indefinite weight, Comm. Partial Differential Equations 15 (1990), 939–982.
M. Faierman, Eigenvalue asymptotics for a non-self adjoint elliptic problem involving an indefinite weight, Rocky Mountain J. Math, (to appear).
M. Faierman, On an oblique derivative problem involving an indefinite weight, Arch. Math. (Brno) (to appear).
M. Faierman, On the eigenvalue asymptotics for a non-self adjoint elliptic problem involving an indefinite weight (submitted).
M. Faierman, On an a priori estimate for solutions of an elliptic equation (submitted).
J. Fleckinger and M.L. Lapidus, Eigenvalues of elliptic boundary value problems with an indefinite weight function, Trans. Amer. Math. Soc. 295 (1986), 305–324.
J. Fleckinger and M.L. Lapidus, Remainder estimates for the asymptotics of elliptic eigenvalue problems with indefinite weights, Arch. Rational Mech. Anal. 98 (1987), 329-356.
. I.C. Gohberg and M.G. Krein, Introduction to the theory of linear nons elf adjoint operators, Amer. Math. Soc., Providence, R.I., 1969.
P. Grisvard, Elliptic problems in nonsmooth domains, Pitman, London, 1985.
P. Hess, On the relative completeness of the generalized eigenvectors of elliptic eigenvalue problems with indefinite weight functions, Math. Ann. 270 (1985), 467–475.
P. Hess, On the asymptotic distribution of eigenvalues of some non-self adjoint problems, Bull. London Math. Soc. 18 (1986), 181–184.
P. Hess, On the spectrum of elliptic operators with respect to indefinite weights, Linear Algebra Appl. 84 (1986), 99–109.
T. Kato, Perturbation theory for linear operators, 2nd edn., Springer, Berlin, 1976.
J.L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, Vol. I, Springer, Berlin, 1972.
A.S. Markus, Introduction to the spectral theory of polynomial operator pencils, Amer. Math. Soc., Providence, R.I., 1988.
V.G. Maz’ja, Sobolev spaces, Springer, Berlin, 1985.
. M. Schechter, General boundary value problems for elliptic partial differential equations, Comm. Pure Appl. Math. 12 (1959), 457–482.
H. Triebel, Interpolation theory, function spaces, differential operators, North- Holland, Amsterdam, 1978.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Birkhäuser Verlag Basel/Switzerland
About this paper
Cite this paper
Faierman, M. (1995). On The Spectral on the Theory of an Elliptic Boundary Value Problem Involving an Indefinite Weight. In: Gohberg, I., Langer, H. (eds) Operator Theory and Boundary Eigenvalue Problems. Operator Theory: Advances and Applications, vol 80. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9106-6_9
Download citation
DOI: https://doi.org/10.1007/978-3-0348-9106-6_9
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0348-9909-3
Online ISBN: 978-3-0348-9106-6
eBook Packages: Springer Book Archive