Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-05T21:40:02.599Z Has data issue: false hasContentIssue false
  • English
  • Français

Positivity results for a nonlocal elliptic equation

Published online by Cambridge University Press:  14 November 2011

Pedro Freitas  and
Guido Sweers
Pedro Freitas
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1096 Lisboa Codex, Portugal
Guido Sweers
Affiliation:
Department of Pure Mathematics, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we consider a second-order linear nonlocal elliptic operator on a bounded domain in ℝn (n ≧ 3), and give conditions which ensure that this operator has a positive inverse. This generalises results of Allegretto and Barabanova, where the kernel of the nonlocal operator was taken to be separable. In particular, our results apply to the case where this kernel is the Green's function associated with second-order uniformly elliptic operators, and thus include the case of some linear elliptic systems. We give several other examples. For a specific case which appears when studying the linearisation of nonlocal parabolic equations around stationary solutions, we also consider the associated eigenvalue problem and give conditions which ensure the existence of a positive eigenfunction associated with the smallest real eigenvalue.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 4 , 1998 , pp. 697 - 715
Copyright
Copyright © Royal Society of Edinburgh 1998

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. and Barabanova, A.. Positivity of solutions of elliptic equations with nonlocal terms. Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), 643–63.Google Scholar
2Ancona, A.. Comparaison des mesures harmoniques et des fonctions de Green pour des opérateurs elliptiques sur un domaine Lipschitzien. C. R. Acad. Sci. Paris 294 (1982), 505–8.Google Scholar
3Bertero, M., Talenti, G. and Viano, G. A.. Scattering and bound state solutions for a class of nonlocal potentials (s-wave). Comm. Math. Phys. 6 (1967), 128–50.CrossRefGoogle Scholar
4Cranston, M., Fabes, E. and Zhao, Z.. Conditional Gauge and potential theory for the Schrödinger operator. Trans. Amer. Math. Soc. 307 (1988), 171–94.Google Scholar
5Davies, E. B.. Heat kernels and spectral theory (Cambridge: Cambridge University Press, 1989).CrossRefGoogle Scholar
6Fiedler, B. and Poláčic, P.. Complicated dynamics of scalar reaction diffusion equations with a nonlocal term. Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), 167–92.CrossRefGoogle Scholar
7Freitas, P.. A nonlocal Sturm-Liouville eigenvalue problem. Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), 169–88.CrossRefGoogle Scholar
8Freitas, P.. Bifurcation and stability of stationary solutions of nonlocal scalar reaction-diffusion equations. J. Dynam. Differential Equations 6 (1994), 613–29.CrossRefGoogle Scholar
9Freitas, P.. Stability of stationary solutions of a nonlocal reaction–diffusion equation. Quart. J. Mech. Appl. Math. 48 (1995), 557–82.CrossRefGoogle Scholar
10Freitas, P. and Vishnevskii, M. P.. Stability of stationary solutions of nonlocal reaction-diffusion equations in m-dimensional space. Differential Integral Equations (to appear).Google Scholar
11Gilbarg, D. and Trudinger, N.. Elliptic partial differential equations of second order, 2nd edn (Berlin: Springer, 1983).Google Scholar
12Grüter, M. and Widman, K.-O.. The Green function for uniformly elliptic equations. Manuscripta Math. 37 (1982), 303–42.CrossRefGoogle Scholar
13Hueber, H. and Sieveking, M.. Uniform bounds for quotients of Green functions on C1,1-domains. Ann. Inst. Fourier 32 (1982), 105–17.Google Scholar
14Liouville, J.. Solution nouvelle d'un problème d'analyse, relatif aux phénomènes thermomécaniques. J. Math. Pures Appl. 2 (1837), 439–56.Google Scholar
15Mitidieri, E. and Sweers, G.. Weakly coupled elliptic systems and positivity. Math. Nachr. 173 (1995), 259–86.CrossRefGoogle Scholar
16Nagel, R., ed., One-parameter semigroups of positive operators, Springer Lecture Notes in Mathematics 1184 (Berlin: Springer, 1986).Google Scholar
17Poláčic, P. and Šošovička, V.. Stable periodic solutions of a spatially homogeneous nonlocal reaction diffusion equation. Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), 867–84.CrossRefGoogle Scholar
18Sweers, G.. Positivity for a strongly coupled elliptic system by Green function estimates. J. Geom. Anal. 4 (1994), 121–42.CrossRefGoogle Scholar
19Zhao, Z.. Green function for Schrödinger operator and conditioned Feynman-Kac gauge. J. Math. Anal. Appl. 116 (1986), 309–34.CrossRefGoogle Scholar