Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-05-18T21:12:53.253Z Has data issue: false hasContentIssue false
  • English
  • Français

An extension of the Sturm–Picone theorem to elliptic differential equations

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

The paper presents an extension of the well-known Sturm–Picone theorem for self-adjoint equations to the n-dimensional case. The basic domain G ⊆ Rn is possibly unbounded and no regularity hypotheses on the boundary ∂G are required. The coefficients of the elliptic equation may also be unbounded on G.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 97 , 1984 , pp. 209 - 215
Copyright
Copyright © Royal Society of Edinburgh 1984

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

1Ahmad, S. and Lazer, A. C.. On the role of Hopf's maximum principle in elliptic Sturmian theory. Houston J. Math. 5 (1979), 155–158.Google Scholar
2Bochenek, J.. On some problems in the theory of eigenvalues and eigenfunctions associated with linear elliptic partial differential equations of the second order. Ann. Polon. Math. 16 (1965), 153–167.CrossRefGoogle Scholar
3Browder, F. E.. On the spectral theory of elliptic differential operators. I. Math. Ann. 142 (1961), 22–130.CrossRefGoogle Scholar
4Chicco, M.. Some properties of the first eigenvalue and the first eigenfunction of linear second order elliptic partial differential equations in divergence form. Bol. Un. Mat. Ital. 5 (1972), 245–254.Google Scholar
5Heywood, J. G., Noussair, E. S. and Swanson, C. A.. On the zeros of solutions of elliptic inequalities in bounded domains. J. Differential Equations 28 (1978), 345–353.CrossRefGoogle Scholar
6Hörmander, L.. Linear partial differential operators (Berlin: Springer, 1963).CrossRefGoogle Scholar
7Kato, T.. Perturbation theory for linear operators (Berlin: Springer, 1966).Google Scholar
8Kinderlehrer, D. and Stampacchia, G.. An Introduction to Variational Inequalities and Their Applications (New York: Academic Press, 1980).Google Scholar
9Krasnoselskii, M. A.. Positive solutions of elliptic equations (Moskau, 1962) (Russian).Google Scholar
10Kreith, K.. Oscillation theory. Lecture Notes in Mathematics 324 (Berlin: Springer, 1973).Google Scholar
11Leighton, W.. Comparison theorems for linear differential equations of second order. Proc. Amer. Math. Soc. 13 (1962), 603–610.CrossRefGoogle Scholar
12McNabb, A.. Strong comparison theorems for elliptic equations of second order. J. Math. Mech. 10 (1961), 431–440.Google Scholar
13Müller-Pfeiffer, E.. Some remarks on Courant's variation principle for elliptic differential operators. Math. Nachr. (in press).Google Scholar
14Swanson, C. A.. Comparison and Oscillation Theory of Linear Differential Equations (New York and London: Academic Press, 1968).Google Scholar