Hostname: page-component-669899f699-2mbcq Total loading time: 0 Render date: 2025-04-27T14:39:53.220Z Has data issue: false hasContentIssue false
  • English
  • Français

Symmetry properties of positive solutions of elliptic equations in an infinite sectorial cone*

Published online by Cambridge University Press:  14 November 2011

Chie-Ping Chu  and
Hwai-Chiuan Wang
Chie-Ping Chu
Affiliation:
Department of Mathematics, Soochow University, Taipei, Taiwan
Hwai-Chiuan Wang
Affiliation:
Department of Mathematics, Tsing Hua University, Hsinchu, Taiwan
Get access Rights & Permissions [Opens in a new window]

Synopsis

We prove symmetry properties of positive solutions of semilinear elliptic equations Δu + f(u) = 0 with Neumann boundary conditions in an infinite sectorial cone. We establish that any positive solution u of such equations in an infinite sectorial cone ∑α in ℝ3 is spherically symmetric if the amplitude α of ∑α is not greater than π.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 122 , Issue 1-2 , 1992 , pp. 137 - 160
Copyright
Copyright © Royal Society of Edinburgh 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1Berestycki, H. and Pacella, F.. Symmetry properties for positive solutions of elliptic equation with mixed boundary conditions. J. Fund. Anal. 87 (1989), 177–211.CrossRefGoogle Scholar
2Bouligand, G.. Sur les founctions de Green et de Neumann du cylindre. Bull. Soc. Math. France 42 (1914), 168–242.CrossRefGoogle Scholar
3Cesare, M. D. and Pacella, F.. Geometrical properties of positive solutions of semilinear elliptic equations in some unbounded domains. Bull. U.M.I. 7 (1989), 691–703.Google Scholar
4Chavel, I.. Eigenvalues in Riemannian Geometry (New York: Academic Press, 1984).Google Scholar
5Cheng, S. Y., and Li, P.. Heat kernel estimates and lower bound of eigenvalues. Comment. Math. Helv. 56 (1981), 327–338.CrossRefGoogle Scholar
6Courant, R. and Hilbert, D.. Methods of Mathematical Physics, Vol. I (New York: Wiley Interscience, 1953).Google Scholar
7Gidas, B., Ni, W. M. and Nirenberg, L.. Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68 (1979), 209–243.CrossRefGoogle Scholar
8Gidas, B., Ni, W. M. and Nirenburg, L.. Symmetry of positive solutions of nonlinear elliptic equations in ℝn. Adv. in Math. Suppl. Stud. 7A (1981), 369–402.Google Scholar
9Gilbarg, D. and Trudinger, N. S.. Elliptic Partial Differential Equations of Second Order, 2nd edn. (Berlin: Springer, Berlin: 1983).Google Scholar
10Li, P. and Yau, S. T.. Estimates of eigenvalues of a compact Riemann manifold. Proc. Sympos. Pure Math. 36 (1980), 205–240.CrossRefGoogle Scholar
11Serrin, J.. A symmetry problem in potential theory. Arch. Rational Mech. Anal. 43 (1971), 304–318.CrossRefGoogle Scholar
12Snow, C.. Hypergeometric and Legendre Functions with Application to Integral Equations of Potential Theory (Washington: National Bureau of Standards Applied Mathematics, Series 19, 1952).Google Scholar