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A general approach to stability in free boundary problems. (English) Zbl 0973.35200
The authors consider the free boundary problem \[ u_t=L u+f, \quad x\in\Omega_t\subset\mathbb{R}^N,\;t>0, \quad u=g_1,\quad \frac{\partial u}{\partial n}=g_2 \quad \text{on }\partial \Omega_t,\tag{1} \] where \(L\) is a uniformly (time independent) elliptic operator and \(f,g_1,g_2\) are given functions. The coefficients of \(L\) and the data are supposed to have sufficient regularity. The authors’ aim is to study the stability of the equilibrium solutions to the problem above, assuming that the corresponding steady state problem is solvable. The main results are: local existence and uniqueness for the evolution problem, the stability theorem, the structure of the stable and unstable manifolds in case of instability (the latter result extending the classical saddle point theorem).
The problem is reformulated in a fixed domain by means of a suitable transformation and linearization in the neighborhood of the equilibrium solution is introduced. The new form taken by (1) is a boundary-value problem for a fully nonlinear parabolic equation with fully nonlinear boundary conditions. The stability analysis is based on the properties of the spectrum of the operator \(L\), which is the realization of \(L\) with homogeneous boundary conditions in \(C(\overline \Omega)\). Some applications are illustrated.

MSC:
35R35 Free boundary problems for PDEs
35B35 Stability in context of PDEs
35K55 Nonlinear parabolic equations
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[1] Berestycki, H.; Caffarelli, L.A.; Nirenberg, L., Uniform estimates for regularization of free boundary problems, (), 567-619 · Zbl 0702.35252
[2] Bertsch, M.; Hilhorst, D.; Schmidt-Lainé, C., The well-posedness of a free-boundary problem arising in combustion theory, Nonl. anal., 23, 1211-1224, (1994) · Zbl 0822.35115
[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, 455-474, (1992) · Zbl 0780.35115
[5] Brauner, C.M.; Lunardi, A.; Schmidt-Lainé, Cl., Multi-dimensional stability analysis of planar travelling waves, Appl. math. letters, 7, 1-4, (1994) · Zbl 0814.35148
[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) · Zbl 0852.35019
[7] Bresch, D.; Simon, J., Sur LES variations normales d’un domaine, ESAIM: control, optimisation and calculus of variations, 3, 251-261, (1998) · Zbl 0907.49021
[8] Bucur, D.; Zolésio, J.P., Shape stability of eigenvalues under capacitary constraints, J. convex anal., 5, 19-30, (1998)
[9] Caffarelli, L.A.; Vazquez, J.L., A free boundary problem for the heat equation arising in flame propagation, Trans. amer. math. soc., 347, 411-441, (1995) · Zbl 0814.35149
[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, 563-608, (1997) · Zbl 0896.35143
[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 · JFM 39.1022.01
[13] Hilhorst, D.; Hulshof, J., An elliptic-parabolic problem in combustion theory: convergence to travelling waves, Nonl. anal., 17, 519-546, (1991) · Zbl 0761.35114
[14] Hilhorst, D.; Hulshof, J., A free boundary focusing problem, Proc. amer. math. soc., 121, 1193-1202, (1994) · Zbl 0805.35164
[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 · Zbl 0557.76001
[17] Lunardi, A., Analytic semigroups and optimal regularity in parabolic problems, (1995), Birkhäuser Basel · Zbl 0816.35001
[18] Simon, J., Differentiation with respect to the domain in boundary value problems, Numer. funct. optimiz., 2, 649-687, (1980) · Zbl 0471.35077
[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, 299-310, (1980) · Zbl 0451.35033
[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 · Zbl 0867.35120
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