Hostname: page-component-54dcc4c588-dbm8p Total loading time: 0 Render date: 2025-09-19T04:42:29.677Z Has data issue: false hasContentIssue false
  • English
  • Français

On the convexity and symmetry of solutions to an elliptic free boundary problem

Part of: Elliptic equations and systems Inequalities Partial differential equations

Published online by Cambridge University Press:  09 April 2009

Bernhard Kawohl
Bernhard Kawohl
Affiliation:
SFB 123, Universität Heidelberg, Im Neuenheimer Feld 294, D6900 Heidelberg, West Germany
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the existence, uniqueness and regularity of solutions to an exterior elliptic free boundary problem. The solutions model stationary solutions to nonlinear diffusion reaction problems, that is, they have compact support and satisfy both homogeneous Dirichlet and Neumann-type boundary conditions on the free boundary ∂{u > 0}. Then we prove convexity and symmetry properties of the free boundary and of the level sets {u > c} of the solutions. We also establish symmetry properties for the corresponding interior free boundary problem.

MSC classification

Secondary: 35J65: Nonlinear boundary value problems for linear elliptic equations 35J20: Variational methods for second-order elliptic equations 35B50: Maximum principles 26D10: Inequalities involving derivatives and differential and integral operators

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 42 , Issue 1 , February 1987 , pp. 57 - 68
Copyright
Copyright © Australian Mathematical Society 1987

References

Bandle, C., Sperb, R. and Stakgold, I. (1984), ‘Diffusion reaction with monotone kinetics’, Nonlinear Anal. 8, 321–333.CrossRefGoogle Scholar
Diaz, J. I. and Herrero, M. A. (1981), ‘Estimates on the support of the solution to some nonlinear elliptic and parabolic problems’, Proc. Roy. Soc. Edinburgh 89 A, 249–258.CrossRefGoogle Scholar
Caffarelli, L. A. and Spruck, J. (1982), ‘Convexity properties of solutions to some classical variational problems’, Comm. Partial Differential Equations 7, 1337–1379.CrossRefGoogle Scholar
Friedman, A. and Phillips, D. (1984), ‘The free boundary of a semilinear elliptic equation’, Trans. Amer. Math. Soc. 282, 153–182.CrossRefGoogle Scholar
Frank, L. S. and Wendt, W. D. (1984), ‘On an elliptic operator with discontinuous nonlinearity’, J. Differential Equations 54, 1–18.CrossRefGoogle Scholar
Gabriel, R. (1957), ‘A result concerning convex level surfaces of 3-dimensional harmonic functions’, J. London Math. Soc. 32, 286–294.CrossRefGoogle Scholar
Gidas, B., Ni, W. M. and Nirenberg, L. (1979), ‘Symmetry and related properties via the maximum principle’, Comm. Math. Phys. 68, 209–243.CrossRefGoogle Scholar
Kawohl, B. (1983 a), ‘Starshapedness of level sets for the obstacle problem and for the capacitary potential problem’, Proc. Amer. Math. Soc. 89, 637–640.CrossRefGoogle Scholar
Kawohl, B. (1983 b), ‘Starshaped rearrangement and applications’ (LCDS Report 83–20, to appear in Trans. Amer. Math. Soc.).Google Scholar
Kawohl, B. (1983 c), ‘Geometrical properties of level sets of solutions to elliptic problems’ (LCDS Report 83–21, to appear in Nonlinear Functional Analysis and Its Applications, Proceedings of Symposia in pure Math., Vol. 45, part 2, 25–36, Amer. Math. Soc., Providence, R. I., 1986.).CrossRefGoogle Scholar
Kawohl, B. (1984 a), ‘On the isoperimetric nature of a rearrangement inequality and its consequences for some variational problems’ (LCDS Report 84–4, to appear in Arch. Ration. Mech. Anal.).Google Scholar
Kawohl, B. (1984 b), ‘More on convex level sets’ (Report 114, Erlangen, appeared in revised form in Comm. Partial Differential Equations 10 (1985), 1213–1225).CrossRefGoogle Scholar
Ladyzhenskaya, O. and Ural'tseva, N. (1968), Linear and quasilinear elliptic equations equations (New York, Academic Press).Google Scholar
Lewis, J. L. (1977), ‘Capacitary functions in convex rings’, Arch. Rational Mech. Analysis 66, 201–224.CrossRefGoogle Scholar
Nečas, J. (1968), Les méthodes directes en théorie des équations elliptiques (Masson, Paris).Google Scholar
Polya, G. and Szegö, G. (1952), Isoperimetric inequalities in mathematical physics (Ann. Math. Studies, Vol. 27, Princeton University Press).Google Scholar