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An unstable elliptic free boundary problem arising in solid combustion. (English) Zbl 1119.35123

The elliptic problem \[ \Delta u=-\chi_{\{u>0\}} \] is unstable in the sense that it corresponds to the classical obstacle problem with inverted sign. It is related to the travelling wave problem for the combustion of a solid with ignition temperature. The main results of the paper are: (i) the existence of a maximal and a minimal solution, (ii) the analyticity of a local minimizer of the energy functional out of the free boundary \(\partial \{u>0\}\) which is also locally analytic, (iii) the analysis of the possible singular points of the free boundary and of the behaviour of \(u\) near them. The two-dimensional case is studied in particular deriving some interesting information on the structure of the free boundary.

MSC:

35R35 Free boundary problems for PDEs
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
35B65 Smoothness and regularity of solutions to PDEs
80A25 Combustion
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[1] N. Aguilera, H. W. Alt, and L. A. Caffarelli, An optimization problem with volume constraint , SIAM J. Control Optim. 24 (1986), 191–198. · Zbl 0588.49005 · doi:10.1137/0324011
[2] A. P. Aldushin and B. I. Khaikin, Combustion of mixtures forming condensed reaction products , Combus. Explos. Shock Waves 10 (1974), 273–280.
[3] H. W. Alt, L. A. Caffarelli, and A. Friedman, Variational problems with two phases and their free boundaries , Trans. Amer. Math. Soc. 282 (1984), 431–461. · Zbl 0844.35137 · doi:10.1090/S0002-9947-1984-0732100-6
[4] J. Andersson and G. S. Weiss, Cross-shaped and degenerate singularities in an unstable elliptic free boundary problem , J. Differential Equations 228 (2006), 633–640. · Zbl 1136.35104 · doi:10.1016/j.jde.2005.11.008
[5] J. M. Beck and V. A. Volpert, Nonlinear dynamics in a simple model of solid flame microstructure , Phys. D 182 (2003), 86–102. · Zbl 1029.80502 · doi:10.1016/S0167-2789(03)00119-2
[6] M.-F. Bidaut-VéRon, V. Galaktionov, P. Grillot, and L. VéRon, Singularities for a \(2\)-dimensional semilinear elliptic equation with a non-Lipschitz nonlinearity , J. Differential Equations 154 (1999), 318–338. · Zbl 0927.35037 · doi:10.1006/jdeq.1998.3567
[7] I. Blank, Eliminating mixed asymptotics in obstacle type free boundary problems , Comm. Partial Differential Equations 29 (2004), 1167–1186. · Zbl 1082.35165 · doi:10.1081/PDE-200033762
[8] L. A. Caffarelli and A. Friedman, The shape of axisymmetric rotating fluid , J. Funct. Anal. 35 (1980), 109–142. · Zbl 0439.35068 · doi:10.1016/0022-1236(80)90082-8
[9] T. Cazenave and P.-L. Lions, Solutions globales d’équations de la chaleur semi linéaires , Comm. Partial Differential Equations 9 (1984), 955–978. · Zbl 0555.35067 · doi:10.1080/03605308408820353
[10] S. Chanillo, D. Grieser, M. Imai, K. Kurata, and I. Ohnishi, Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes , Comm. Math. Phys. 214 (2000), 315–337. · Zbl 0972.49030 · doi:10.1007/PL00005534
[11] S. Chanillo, D. Grieser, and K. Kurata, “The free boundary problem in the optimization of composite membranes” in Differential Geometric Methods in the Control of Partial Differential Equations (Boulder, Colo., 1999) , Contemp. Math. 268 , Amer. Math. Soc., Providence, 2000, 61–81. · Zbl 0988.35124
[12] J.-Y. Chemin, Perfect Incompressible Fluids , Oxford Lecture Ser. Math. Appl. 14 , Oxford Univ. Press, New York, 1998.
[13] X.-Y. Chen, Uniqueness of the \(\omega\) -limit point of solutions of a semilinear heat equation on the circle, Proc. Japan Acad. Ser. A Math. Sci. 62 (1986), 335–337. · Zbl 0641.35028 · doi:10.3792/pjaa.62.335
[14] X.-Y. Chen, H. Matano, and L. VéRon, Anisotropic singularities of solutions of nonlinear elliptic equations in \(\mathbf R^ 2\), J. Funct. Anal. 83 (1989), 50–97. · Zbl 0687.35020 · doi:10.1016/0022-1236(89)90031-1
[15] R. Gianni and J. Hulshof, The semilinear heat equation with a Heaviside source term , European J. Appl. Math. 3 (1992), 367–379. · Zbl 0789.35088 · doi:10.1017/S0956792500000917
[16] E. Giusti, Minimal Surfaces and Functions of Bounded Variation , Monogr. Math. 80 , Birkhäuser, Basel, 1984. · Zbl 0545.49018
[17] D. Kinderlehrer, L. Nirenberg, and J. Spruck, Regularity in elliptic free boundary problems , J. Analyse Math. 34 (1978), 86–119. · Zbl 0402.35045 · doi:10.1007/BF02790009
[18] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications , Pure Appl. Math. 88 , Academic Press, New York, 1980. · Zbl 0457.35001
[19] V. A. Knyazik, A. G. Merzhanov, V. B. Solomonov, and A. S. Shteinberg, Macrokinetics of high-temperature titanium interaction with carbon under electrothermal explosion conditions , Combus. Explos. Shock Waves 21 (1985), 69–73.
[20] J. Norbury and A. M. Stuart, A model for porous-medium combustion , Quart. J. Mech. Appl. Math. 42 (1989), 159–178. · Zbl 0679.76104 · doi:10.1093/qjmam/42.1.159
[21] H. Shahgholian, The singular set for the composite membrane problem , to appear in Comm. Math. Phys., preprint, 2005. · Zbl 1157.35125 · doi:10.1007/s00220-006-0160-8
[22] H. Shahgholian, N. Uraltseva, and G. S. Weiss, Global solutions of an obstacle-problem-like equation with two phases , Monatsh. Math. 142 (2004), 27–34. · Zbl 1057.35098 · doi:10.1007/s00605-004-0235-6
[23] A. Varma, A. S. Mukasyan, and S. Hwang, Dynamics of self-propagating reactions in heterogeneous media: Experiments and model , Chem. Eng. Sci. 56 (2001), 1459–1466.
[24] G. S. Weiss, Partial regularity for weak solutions of an elliptic free boundary problem , Comm. Partial Differential Equations 23 (1998), 439–455. · Zbl 0897.35017 · doi:10.1080/03605309808821352
[25] -, Partial regularity for a minimum problem with free boundary , J. Geom. Anal. 9 (1999), 317–326. · Zbl 0960.49026 · doi:10.1007/BF02921941
[26] -, An obstacle-problem-like equation with two phases: Pointwise regularity of the solution and an estimate of the Hausdorff dimension of the free boundary , Interfaces Free Bound. 3 (2001), 121–128. · Zbl 0986.35139 · doi:10.4171/IFB/35
[27] O. Zik, Z. Olami, and E. Moses, Fingering instability in combustion , Phys. Rev. Lett. 81 (1998), 3868–3871.
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