zbMATH — the first resource for mathematics

The free boundary of a semilinear elliptic equation. (English) Zbl 0552.35079
This paper contains a detailed study of the Dirichlet problem $$\Delta u=\lambda f(u)$$ in $$\Omega$$, $$(\lambda >0)$$; $$u=1$$ on $$\partial \Omega$$, where f:$${\mathbb{R}}\to {\mathbb{R}}$$ is a function satisfying $$f(t)=0$$, if $$t\geq 0$$, $$f(t)>0$$, if $$t>0$$, and $$f(t)\sim t^ p$$ as $$t\downarrow 0$$, $$0<p<1$$. Special emphasis is layed on the study of the ”dead core” $$N_{\lambda}=\{u_{\lambda}=0\}$$ and the properties of the ”free boundary” $$\partial N_{\lambda}$$. For instance the authors prove that for $$\lambda$$ large $$\partial N_{\lambda}$$ is a smooth surface parallel to $$\partial \Omega$$ at a distance $$\gamma /\sqrt{\lambda}+O(1/\lambda),$$ $$\gamma$$ constant; for convex domains $$\Omega \subset {\mathbb{R}}^ 2$$ and for suitable f, $$N_{\lambda}$$ is shown to be convex. Most of the results are extended to the case of the Rodin problem that is the boundary condition changes to $$\partial u/\partial \nu +\mu (u+1)=0$$ on $$\partial \Omega$$.
Reviewer: R.Landes

MSC:
 35R35 Free boundary problems for PDEs 35J65 Nonlinear boundary value problems for linear elliptic equations 35J20 Variational methods for second-order elliptic equations
Full Text:
References:
 [1] H. W. Alt and D. Phillips, A free boundary problem for semilinear elliptic equations, J. Reine Angew. Math. 368 (1986), 63 – 107. · Zbl 0598.35132 [2] R. Aris, The mathematical theory of diffusion and reaction in permeable catalysts, Clarendon Press, Oxford, 1975. · Zbl 0315.76051 [3] C. Bandle, R. P. Sperb and I. Stakgold, The single steady state irreverisble reaction (to appear). · Zbl 0545.35011 [4] Luis A. Caffarelli and Joel Spruck, Convexity properties of solutions to some classical variational problems, Comm. Partial Differential Equations 7 (1982), no. 11, 1337 – 1379. · Zbl 0508.49013 · doi:10.1080/03605308208820254 · doi.org [5] Donald S. Cohen and Theodore W. Laetsch, Nonlinear boundary value problems suggested by chemical reactor theory., J. Differential Equations 7 (1970), 217 – 226. · Zbl 0201.43102 · doi:10.1016/0022-0396(70)90106-3 · doi.org [6] R. Courant and D. Hilbert, Methods of mathematical physics, II: Partial differential equations, Interscience, New York, 1962. · Zbl 0099.29504 [7] J. Ildefonso Diaz and Jesús Hernández, On the existence of a free boundary for a class of reaction-diffusion systems, SIAM J. Math. Anal. 15 (1984), no. 4, 670 – 685. · Zbl 0556.35126 · doi:10.1137/0515052 · doi.org [8] Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. · Zbl 0176.00801 [9] Enrico Giusti, Minimal surfaces and functions of bounded variation, Department of Pure Mathematics, Australian National University, Canberra, 1977. With notes by Graham H. Williams; Notes on Pure Mathematics, 10. · Zbl 0402.49033 [10] Herbert B. Keller, Elliptic boundary value problems suggested by nonlinear diffusion processes, Arch. Rational Mech. Anal. 35 (1969), 363 – 381. · Zbl 0188.17102 · doi:10.1007/BF00247683 · doi.org [11] Hans Lewy and Guido Stampacchia, On the regularity of the solution of a variational inequality, Comm. Pure Appl. Math. 22 (1969), 153 – 188. · Zbl 0167.11501 · doi:10.1002/cpa.3160220203 · doi.org [12] J. Mossino, A priori estimates for a model of Grad Mercier type in plasma confinement, Applicable Anal. 13 (1982), no. 3, 185 – 207. · Zbl 0478.35018 · doi:10.1080/00036818208839390 · doi.org [13] L. E. Payne and G. A. Philippin, Some maximum principles for nonlinear elliptic equations in divergence form with applications to capillary surfaces and to surfaces of constant mean curvature, Nonlinear Anal. 3 (1979), no. 2, 193 – 211. · Zbl 0408.35015 · doi:10.1016/0362-546X(79)90076-2 · doi.org [14] L. E. Payne and I. Stakgold, On the mean value of the fundamental mode in the fixed membrane problem, Applicable Anal. 3 (1973), 295 – 306. Collection of articles dedicated to Alexander Weinstein on the occasion of his 75th birthday. · Zbl 0323.35057 · doi:10.1080/00036817308839071 · doi.org [15] Daniel Phillips, A minimization problem and the regularity of solutions in the presence of a free boundary, Indiana Univ. Math. J. 32 (1983), no. 1, 1 – 17. · Zbl 0545.35013 · doi:10.1512/iumj.1983.32.32001 · doi.org [16] Daniel Phillips, Hausdorff measure estimates of a free boundary for a minimum problem, Comm. Partial Differential Equations 8 (1983), no. 13, 1409 – 1454. · Zbl 0555.35128 · doi:10.1080/03605308308820309 · doi.org [17] R. Sperb and I. Stakgold, Estimates for membranes of varying density, Applicable Anal. 8 (1978/79), no. 4, 301 – 318. · Zbl 0402.35034 · doi:10.1080/00036817908839240 · doi.org [18] I. Stakgold, Gradient bounds for plasma confinement, Math. Methods Appl. Sci. 2 (1980), no. 1, 68 – 72. · Zbl 0431.35087 · doi:10.1002/mma.1670020107 · doi.org [19] -, Estimates for some free bounary problems, Ordinary and Partial Differential Equations, Lecture Notes in Math., vol. 846, Springer-Verlag, Berlin and New York, 1982.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.