Starshapedness of level sets for the obstacle problem and for the capacitory potential problem. (English) Zbl 0556.35051

The author proves that if the geometry of the data in the so-called ’obstacle’ problem and the ’capacitory potential’ problem is starshaped, then so are the solutions. The proofs are based on appropriate maximum principles.
Reviewer: A.D.Osborne


35J65 Nonlinear boundary value problems for linear elliptic equations
35A30 Geometric theory, characteristics, transformations in context of PDEs
35B50 Maximum principles in context of PDEs
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[1] Christer Borell, Brownian motion in a convex ring and quasiconcavity, Comm. Math. Phys. 86 (1982), no. 1, 143 – 147. · Zbl 0516.60084
[2] Haïm Brézis, Problèmes unilatéraux, J. Math. Pures Appl. (9) 51 (1972), 1 – 168. · Zbl 0221.35028
[3] 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
[4] Avner Friedman, Variational principles and free-boundary problems, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982. A Wiley-Interscience Publication. · Zbl 0564.49002
[5] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin-New York, 1977. Grundlehren der Mathematischen Wissenschaften, Vol. 224. · Zbl 0361.35003
[6] David Kinderlehrer, Variational inequalities and free boundary problems, Bull. Amer. Math. Soc. 84 (1978), no. 1, 7 – 26. · Zbl 0382.35004
[7] David Kinderlehrer and Guido Stampacchia, An introduction to variational inequalities and their applications, Pure and Applied Mathematics, vol. 88, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. · Zbl 0457.35001
[8] G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, no. 27, Princeton University Press, Princeton, N. J., 1951. · Zbl 0044.38301
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