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Radial symmetry and uniqueness for an overdetermined problem
Author(s) -
Greco Antonio
Publication year - 2001
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/1099-1476(20010125)24:2<103::aid-mma200>3.0.co;2-f
Subject(s) - overdetermined system , mathematics , uniqueness , symmetry in biology , mathematical analysis , domain (mathematical analysis) , boundary (topology) , constant (computer programming) , symmetry (geometry) , boundary value problem , function (biology) , uniqueness theorem for poisson's equation , pure mathematics , geometry , evolutionary biology , computer science , biology , programming language
Consider a function u , harmonic in a ring‐shaped domain and taking two constant (distinct) values on the two connected components of the boundary. If we know in advance that one of the components is a sphere, and that u satisfies some overdetermined condition on the other one, can we conclude that u is radial? This paper answers this question for certain overdetermined conditions on the gradient of u , generalizing some previous results. Conditions depending on the principal curvatures of the boundary are also investigated. Existence and uniqueness of a radial solution to the overdetermined problem are discussed. Some extensions to ellipsoidal domains, as well as to quasilinear elliptic equations, are carried out. Copyright © 2001 John Wiley & Sons, Ltd.