Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-05-16T05:24:38.024Z Has data issue: false hasContentIssue false
  • English
  • Français

On the existence of nodal domains for elliptic differential operators

Published online by Cambridge University Press:  14 November 2011

E. Müller-Pfeiffer
E. Müller-Pfeiffer
Affiliation:
Erfurt, G.D.R
Get access Rights & Permissions [Opens in a new window]

Synopsis

We prove that the selfadjoint elliptic differential equation (1) has rectangular nodal domains if the quadratic form of the equation takes on negative values. The existence of nodal domains is closely connected with the position of the smallest point of the spectrum of the corresponding selfadjoint operator (Friedrichs extension). If the smallest point of a second order selfadjoint differential operator with Dirichlet boundary conditions is an eigenvalue, then this eigenvalue is strictly increasing when the (possibly unbounded) domain, where the coefficients of the differential operator are denned, is shrinking (Theorem 4).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 94 , Issue 3-4 , 1983 , pp. 287 - 299
Copyright
Copyright © Royal Society of Edinburgh 1983

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Allegretto, W.. On the equivalence of two types of oscillation for elliptic operators. Pacific J. Math. 55 (1974), 319–328.CrossRefGoogle Scholar
2Browder, F. E.. On the spectral theory of elliptic differential operators I. Math. Ann. 142 (1961), 22–130.CrossRefGoogle Scholar
3Chicco, M.. Some properties of the first eigenvalue and the first eigenfunction of linear second order elliptic partial differential equations in divergence form. Boll. Un. Mat. Ital. 5 (1972), 245–254.Google Scholar
4Courant, R. and Hilbert, D.. Methoden der mathematischen Physik I (Berlin: Springer, 1968).CrossRefGoogle Scholar
5Dunford, N. and Schwartz, J. T.. Linear Operators, Part II (New York Interscience, 1963).Google Scholar
6Etgen, G. J. and Lewis, R. T.. The oscillation of elliptic systems. Math. Nachr. 94 (1980), 43–50.CrossRefGoogle Scholar
7Glazman, I. M.. Direct methods of qualitative spectral analysis of singular differential operators (Jerusalem: I.P.S.T., 1965).Google Scholar
8Hörmander, L., Linear partial differential operators (Berlin: Springer, 1963).CrossRefGoogle Scholar
9Kato, T.. Perturbation theory for linear operators (Berlin: Springer, 1966).Google Scholar
10Krasnoselskii, M. A.. Positive solutions of operator equations (Moscow 1962) (in Russian).Google Scholar
11Moss, W. F. and Piepenbrink, J.. Positive solutions of ellipitic equations. Pacific J. Math. 75 (1978), 219–226.CrossRefGoogle Scholar
12Reid, W. T.. Ricatti matrix differential equations and non-oscillation criteria for associated systems, Pacific J. Math. 13 (1963), 665–685.CrossRefGoogle Scholar
13Schmincke, U.-W.. The lower spectrum of Schrödinger operators. Arch. Rational Mech. Anal. 75 (1981), 147–155.CrossRefGoogle Scholar