The free boundary of a semilinear elliptic equation
HTML articles powered by AMS MathViewer
- by Avner Friedman and Daniel Phillips PDF
- Trans. Amer. Math. Soc. 282 (1984), 153-182 Request permission
Abstract:
The Dirichlet problem $\Delta u = \lambda f(u)$ in a domain $\Omega , u = 1$ on $\partial \Omega$ is considered with $f(t) = 0$ if $t \leq 0, f(t) > 0$ if $t > 0, f(t) \sim {t^p}$ if $t \downarrow 0,0 < p < 1;f(t)$ is not monotone in general. The set $\{ u = 0\}$ and the “free boundary” $\partial \{ u = 0\}$ are studied. Sharp asymptotic estimates are established as $\lambda \to \infty$. For suitable $f$, under the assumption that $\Omega$ is a two-dimensional convex domain, it is shown that $\{ u = 0\}$ is a convex set. Analogous results are established also in the case where $\partial u/\partial v + \mu (u - 1) = 0$ on $\partial \Omega$.References
- H. W. Alt and D. Phillips, A free boundary problem for semilinear elliptic equations, J. Reine Angew. Math. 368 (1986), 63–107. MR 850615 R. Aris, The mathematical theory of diffusion and reaction in permeable catalysts, Clarendon Press, Oxford, 1975. C. Bandle, R. P. Sperb and I. Stakgold, The single steady state irreverisble reaction (to appear).
- 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. MR 678504, DOI 10.1080/03605308208820254
- Donald S. Cohen and Theodore W. Laetsch, Nonlinear boundary value problems suggested by chemical reactor theory, J. Differential Equations 7 (1970), 217–226. MR 259356, DOI 10.1016/0022-0396(70)90106-3 R. Courant and D. Hilbert, Methods of mathematical physics, II: Partial differential equations, Interscience, New York, 1962.
- 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. MR 747428, DOI 10.1137/0515052
- Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
- Enrico Giusti, Minimal surfaces and functions of bounded variation, Notes on Pure Mathematics, vol. 10, Australian National University, Department of Pure Mathematics, Canberra, 1977. With notes by Graham H. Williams. MR 0638362
- Herbert B. Keller, Elliptic boundary value problems suggested by nonlinear diffusion processes, Arch. Rational Mech. Anal. 35 (1969), 363–381. MR 255979, DOI 10.1007/BF00247683
- Hans Lewy and Guido Stampacchia, On the regularity of the solution of a variational inequality, Comm. Pure Appl. Math. 22 (1969), 153–188. MR 247551, DOI 10.1002/cpa.3160220203
- J. Mossino, A priori estimates for a model of Grad Mercier type in plasma confinement, Applicable Anal. 13 (1982), no. 3, 185–207. MR 663773, DOI 10.1080/00036818208839390
- 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. MR 525971, DOI 10.1016/0362-546X(79)90076-2
- 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. MR 399633, DOI 10.1080/00036817308839071
- 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. MR 684751, DOI 10.1512/iumj.1983.32.32001
- Daniel Phillips, Hausdorff measure estimates of a free boundary for a minimum problem, Comm. Partial Differential Equations 8 (1983), no. 13, 1409–1454. MR 714047, DOI 10.1080/03605308308820309
- R. Sperb and I. Stakgold, Estimates for membranes of varying density, Applicable Anal. 8 (1978/79), no. 4, 301–318. MR 530105, DOI 10.1080/00036817908839240
- I. Stakgold, Gradient bounds for plasma confinement, Math. Methods Appl. Sci. 2 (1980), no. 1, 68–72. MR 561379, DOI 10.1002/mma.1670020107 —, Estimates for some free bounary problems, Ordinary and Partial Differential Equations, Lecture Notes in Math., vol. 846, Springer-Verlag, Berlin and New York, 1982.
Additional Information
- © Copyright 1984 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 282 (1984), 153-182
- MSC: Primary 35J65; Secondary 35J85, 35R35
- DOI: https://doi.org/10.1090/S0002-9947-1984-0728708-4
- MathSciNet review: 728708