Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-26T20:19:21.011Z Has data issue: false hasContentIssue false
  • English
  • Français

A generic property of the resonance set of an elliptic operator with respect to the domain*

Published online by Cambridge University Press:  14 November 2011

Angela Pistoia
Angela Pistoia
Affiliation:
Dipartimento di Matematica Applicata ‘U Dini’, Facoltá di Ingegneria, via Bonanno 25, 56100 Pisa, Italy email: pistoia@mail.dm.unipi.it
Get access Rights & Permissions [Opens in a new window]

Extract

The set of points (α, β) ∈ℝ2 for which the problem − δu = αu+ − βu− in Ω, u = 0 on ∂Ω(u+ = max {u, o} and u− = min {−u, 0}) has nontrivial solutions is important for the study of certain nonlinear problems. It is shown that for ‘most’ bounded domains Ω in ℝn, such a set is locally the union of a finite number of curves.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 127 , Issue 6 , 1997 , pp. 1301 - 1310
Copyright
Copyright © Royal Society of Edinburgh 1997

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

1Aronszajn, N., Krzywicki, A. and Szarski, I.. A unique continuation theorem for exterior differential forms on Riemannian manifolds. Ark. Mat. 4 (1962), 417–53.CrossRefGoogle Scholar
2Caffarelli, L. A. and Friedman, A.. Partial regularity of the zero set of solutions of linear and superlinear elliptic equations. J. Differential Equations 60 (1985), 420–33.CrossRefGoogle Scholar
3Cartan, H.. Calcul differentiel (Paris: Hermann, 1967).Google Scholar
4Dancer, E. N.. On the Dirichlet problems for weakly nonlinear elliptic partial differential equations. Proc. Roy. Soc. Edinburgh Sect. A 76 (1977), 283–300.CrossRefGoogle Scholar
5Dancer, E. N.. On the existence of solutions of certain asymptotically homogeneous problems. Math. Z. 177 (1981), 33–48.CrossRefGoogle Scholar
6Dancer, E. N.. Generic domain dependence for non-smooth equations and the open set problem for jumping nonlinearities. Topol. Methods Nonlinear Anal. 1 (1993), 139–50.CrossRefGoogle Scholar
7Dancer, E. N. and Du, Y.. Competing species equations with diffusion, large interactions and jumping nonlinearities. J. Differential Equations 114 (1994), 434–75.CrossRefGoogle Scholar
8D'Aujourd'hui, M.. The stability of the resonance set for a problem with jumping nonlinearities. Proc. Roy. Soc. Edinburgh Sect. A 107 (1987), 201–12.CrossRefGoogle Scholar
9De Figueiredo, D. G. and Gossez, J. P.. On the first curve of the Fucik spectrum of an elliptic operator (Preprint).Google Scholar
10Fucik, S.. Solvability of nonlinear equations and boundary value problems (Dordrecht: Reidel, 1990).Google Scholar
11Gallouet, T. and Kavian, O.. Resultats d'existence et nonexistence pour certains problèmes dèmilineaires à l'infmi. Ann. Fac. Sci. Toulouse Math. 3 (1981), 201–46.CrossRefGoogle Scholar
12Giannoni, F. and Micheletti, A. M.. Some remarks about elliptic problems with jumping nonlinearity. Rend. Mat. 7 (1987), 145–57.Google Scholar
13Ladyzenskaja, O. A. and Uralceva, N. N.. Equations aux dèrivèes partielles de type elliptique (Paris: Dunod, 1968).Google Scholar
14Landis, E.. On some properties of solutions of elliptic equations. Dokl. Akad. Nauk. SSSR 107 (1956), 640–3.Google Scholar
15Magalhaes, C. A.. Semilinear elliptic problem with crossing of multiple eigenvalues. Comm. Partial Differential Equations 15(9) (1990), 1265–92.CrossRefGoogle Scholar
16Micheletti, A. M.. Perturbazione dello spettro dell'operatore di Laplace in relazione ad una variazione del campo. Ann. Scuola Norm. Sup. Pisa 26 (1972), 151–69.Google Scholar
17Micheletti, A. M.. A remark on the resonance set for a semilinear elliptic equation. Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), 803–09.CrossRefGoogle Scholar
18Micheletti, A. M. and Pistoia, A.. A note on the resonance set for a semilinear elliptic equation and an application to jumping nonlinearities. Topol. Methods Nonlinear Anal. 6 (1995), 67–80.CrossRefGoogle Scholar
19Murat, F. and Simon, J.. Sur le controle par un domaine gèometrique (Publ. du Lab. Ass. No. 189, Univ. Paris VI, No. 76105, 1976).Google Scholar
20Ruf, B.. On nonlinear elliptic problems with jumping nonlinearities. Ann. Mat. Pura Appl. 128 (1980), 133–51.CrossRefGoogle Scholar
21Quinn, F.. Transversal approximation on Banach manifolds. Global Analysis, Vol. 111, pp. 213–22 (Providence, RI: American Mathematical Society, 1970).CrossRefGoogle Scholar
22Saut, J. and Temam, R.. Generic properties of nonlinear boundary value problems. Comm. Partial Differential Equations 4 (1979), 293–319.CrossRefGoogle Scholar