Starshapedness of level sets for the obstacle problem and for the capacitory potential problem

Kawohl, Bernhard

doi:10.1090/S0002-9939-1983-0718988-8

Starshapedness of level sets for the obstacle problem and for the capacitory potential problem
HTML articles powered by AMS MathViewer

by Bernhard Kawohl PDF

Proc. Amer. Math. Soc. 89 (1983), 637-640 Request permission

Abstract:

If the geometry of the data in the obstacle problem and the capacitory potential problem is starshaped then so are the solutions. The proofs are based on appropriate maximum principles.

References

Christer Borell, Brownian motion in a convex ring and quasiconcavity, Comm. Math. Phys. 86 (1982), no. 1, 143–147. MR 678006
Haïm Brézis, Problèmes unilatéraux, J. Math. Pures Appl. (9) 51 (1972), 1–168. MR 428137
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
Avner Friedman, Variational principles and free-boundary problems, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1982. MR 679313
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977. MR 0473443
David Kinderlehrer, Variational inequalities and free boundary problems, Bull. Amer. Math. Soc. 84 (1978), no. 1, 7–26. MR 466885, DOI 10.1090/S0002-9904-1978-14397-3
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. MR 567696
G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, No. 27, Princeton University Press, Princeton, N. J., 1951. MR 0043486