A General Approach to Stability in Free Boundary Problems

Journal of Differential Equations

Volumn 164, Issue 1, 2000, Pages 16-48

  Brauner, Claude Michel ^a   Hulshof, Josephus ^b   Lunardi, Alessandra ^c  

^a UNIVERSITÉ DE BORDEAUX   (France)

^b LEIDEN UNIVERSITY   (Netherlands)

^c UNIVERSITY OF PARMA   (Italy)

Author keywords

Free boundary problems; focussing selfsimilar solutions; linearisation; analytic semigroups; fully nonlinear equations; stable and unstable manifolds; saddle point property

Indexed keywords

EID: 0002404790     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1006/jdeq.1999.3734     Document Type: Article

References (21)

1. Berestycki H., Caffarelli L. A., Nirenberg L. Uniform Estimates for Regularization of Free Boundary Problems. Sadosky C. Analysis and Partial Differential Equations. 1990;567-619 Marcel Dekker.
2. Bertsch M., Hilhorst D., Schmidt-Lainé C. The well-posedness of a free-boundary problem arising in combustion theory. Nonl. Anal. 23:1994;1211-1224.
3. C. M. Brauner, J. Hulshof, and Cl. Schmidt-Lainé, The saddle point property for focusing selfsimilar solutions of a combustion type free boundary problem: the radial case, Proc. AMS1271999, 473-479, 1999.
4. Brauner C. M., Lunardi A., Schmidt-Lainé Cl. Stability of travelling waves with interface conditions. Nonl. Anal. 19:1992;455-474.
5. Brauner C. M., Lunardi A., Schmidt-Lainé Cl. Multi-dimensional stability analysis of planar travelling waves. Appl. Math. Letters. 7:1994;1-4.
6. Brauner C. M., Lunardi A., Schmidt-Lainé Cl. Stability of Travelling Waves in a Multidimensional Free Boundary Problem. Scuola Norm. Sup. Pisa 3. 1996.
7. Bresch D., Simon J. Sur les variations normales d'un domaine. ESAIM: Control, Optimisation and Calculus of Variations. 3:1998;251-261.
8. Bucur D., Zolésio J. P. Shape stability of eigenvalues under capacitary constraints. J. Convex Anal. 5:1998;19-30.
9. Caffarelli L. A., Vazquez J. L. A free boundary problem for the heat equation arising in flame propagation. Trans. Amer. Math. Soc. 347:1995;411-441.
10. Dervieux A. Perturbation des équations d'équilibre d'un plasma confiné: comportement de la frontière libre, étude des branches de solutions. INRIA Research Report. 18:1980.
11. Galaktionov V. A., Hulshof J., Vazquez J. L. Extinction and focusing behaviour of spherical and annular flames described by a free boundary problem. J. Math. Pures Appl. 76:1997;563-608.
12. Hadamard J. Mémoire sur le problème d'analyse relatif à l'équilibre des plaques élastiques encastrées. Oeuvres de J. Hadamard. 1968;CNRS, Paris. p. 1907.
13. Hilhorst D., Hulshof J. An elliptic-parabolic problem in combustion theory: convergence to travelling waves. Nonl. Anal. 17:1991;519-546.
14. Hilhorst D., Hulshof J. A free boundary focusing problem. Proc. Amer. Math. Soc. 121:1994;1193-1202.
15. Ladyzhenskaja O. A., Solonnikov V. A., Ural'ceva N. N. Linear and quasilinear equations of parabolic type. 1967;Nauka, Moskow.
16. Buckmaster J. D., Ludford G. S. S. Theory of Laminar Flames. 1982;Cambridge University Press, Cambridge.
17. Lunardi A. Analytic Semigroups and Optimal Regularity in Parabolic Problems. 1995;Birkhäuser, Basel.
18. Simon J. Differentiation with respect to the domain in boundary value problems. Numer. Funct. Optimiz. 2:1980;649-687.
19. Simon J. Optimal Design for Neumann Condition and for Related Boundary Value Conditions. Publications du Laboratoire d'Analyse Numérique. 1986;Université Paris 6.
20. Stewart H. B. Generation of analytic semigroups by strongly elliptic operators under general boundary conditions. Trans. Amer. Math. Soc. 259:1980;299-310.
21. Vazquez J. L. The free boundary problem for the heat equation with fixed gradient condition. Proc. Int. Conf. 'Free Boundary Problems, Theory and Applications,' Zakopane, Poland. Pitman Research Notes in Mathematics. 363:1995;. p. 277-302.