Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-23T22:19:43.124Z Has data issue: false hasContentIssue false
  • English
  • Français

Padé approximant bounds on the positive solutions of some nonlinear elliptic equations

Published online by Cambridge University Press:  14 November 2011

M. F. Barnsley  and
D. Bessis
M. F. Barnsley
Affiliation:
School of Mathematics, University of Bradford
D. Bessis
Affiliation:
Service de Physique Théorique, C.E.N.S., B.P. No. 2, 91190 Gif-sur-Yvette, France
Get access Rights & Permissions [Opens in a new window]

Synopsis

We consider the equation Lφ − λpφ + γqφ2 = f on a bounded domain in Rn with homogeneous Neumann-Dirichlet boundary conditions. L is a negative definite uniformly elliptic differential operator, while, p, q and f are positive functions. We show that there exists exactly one positive solution for each λ ∈ R and γ > 0. This solution can be analytically continued throughout Re γ > 0: it is a Laplace transform of a positive measure. The measure is bounded prior to the bifurcation point of the associated “homogeneous” equation and unbounded after. Noting that any Laplace transform of positive measure has associated with it a natural sequence of Tchebycheff systems, it now follows that one can obtain monotonically converging upper and lower bounds which are provided by the generalized Padé approximants generated from the Tchebycheff systems.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 80 , Issue 3-4 , 1978 , pp. 183 - 200
Copyright
Copyright © Royal Society of Edinburgh 1978

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

1Miranda, C.Partial Differential Equations of Elliptic Type, 2nd edn revd (Berlin: Springer 1970).Google Scholar
2Protter, M. H. and Weinberger, H. F.Maximum Principles in Differential Equations (Englewood Cliffs, N.J.: Prentice-Hall, 1967).Google Scholar
3Agmon, S., Douglis, A. and Nirenberg, L.Estimates Near the Boundary of Elliptic Partial differential Equations Satisfying General Boundary Conditions I. Comm. Pure Appl. Math. 12 (1959), 623–727.CrossRefGoogle Scholar
4Sattinger, D. H.Monotone Methods in Nonlinear Elliptic and Parabolic Boundary Value Problems. Indiana Univ. Math. J. 21 (1972), 979–1000.CrossRefGoogle Scholar
5Bessis, D., Moussa, P. and Villani, M., Monotonic converging variational approximations to the functional integrals in quantum statistical mechanics. J. Mathematical Phys. 16 (1975) 2318–2325.CrossRefGoogle Scholar
6Barnsley, M. F.Sum rule functions. J. Mathematical Phys. 12 (1971), 957–964.CrossRefGoogle Scholar
7Keller, H. B. and Cohen, D. S.Some Positone Problems Suggested by Nonlinear Heat Generation. J. Math. Mech. 16 (1967), 1361–1376.Google Scholar
8Amann, H.On the Existence of Positive Solutions of Nonlinear Elliptic and Parabolic Boundary Value Problems. Indiana Univ. Math. J. 21 (1971), 125–146.CrossRefGoogle Scholar
9Widder, D. V.The Laplace Transform (Princeton: Univ. Press, 1941).Google Scholar
10Stakgold, I. and Payne, L. E. Nonlinear Problems in Nuclear Reactor Analysis. In Nonlinear Problems in the Physical Sciences and Biology. Lecture Notes in Mathematics 332 (Berlin: Springer, 1973).CrossRefGoogle Scholar
11Baker, G. A. Jr, The Padé Approximant Method and some Related Generalizations. In The Padé Approximant in Theoretical Physics (Eds. Baker, G. A. Jr and Gammel, J. L.) (New York: Academic Press, 1970).Google Scholar
12Baker, G. A. Jr. The theory and Application of the Padé Approximant Method. Advances in Theoretical Physics I (New York: Academic Press, 1965).Google Scholar
13Baker, G. A. Jr, Best Error Bounds for Padé Approximants to Convergent Series of Stieltjes. J. Mathematical Phys. 10 (1969), 814–820.CrossRefGoogle Scholar
14Barnsley, M. F.The Bounding Properties of the Multipoint Padé Approximant to a Series of Stieltjes on the Real Line. J. Mathematical Phys. 14 (1974), 299–313.CrossRefGoogle Scholar
15Cohen, D. S.Positive Solutions of Nonlinear Eigenvalue Problems: Applications to Nonlinear Reactor Dynamics. Arch. Rational Mech. Anal. 26 (1968), 305–315.CrossRefGoogle Scholar
16Shohat, J. A. and Tamarkin, J. D. The Problem of Moments. Math. Surveys 1 (Providence, R.I.: Amer. Math. Soc, 1950).Google Scholar
17Karlin, S. and Studden, W. J.Tchebycheff systems (New York: Interscience, 1966).Google Scholar