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A free boundary problem in an annulus

Part of: Two-dimensional theory

Published online by Cambridge University Press:  09 April 2009

David E. Tepper
David E. Tepper
Affiliation:
Department of Mathematics Baruch College City University of New YorkNew York, New York 10010, U.S.A.
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Abstract

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If Ω is a ring region with starlike boundary components α and β, then we show for each λ > 0 there exists a ring region ω ⊂ Ω with ∂ ω = α ∪ ϒ, α ∩ ϒ = φ such that there is a harmonic function V in ω satisfying (a) V(z) = 0 for z ∈ α, (b) V(z) = 1 for z ∈ ϒ, (c) | grad V(z)| = λ for z ∈ ϒ ∩ Ω. Furthermore, we show when ω is not equal to Ω; that is, that is, there is a non-trival solution.

MSC classification

Secondary: 31A05: Harmonic, subharmonic, superharmonic functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 34 , Issue 2 , April 1983 , pp. 177 - 181
Copyright
Copyright © Australian Mathematical Society 1983

References

1.Acker, A., ‘Heat flow inequalities with applications to heat flow optimization problem’, SIAM J. Math. Anal. 8 (1977), 504–18.CrossRefGoogle Scholar
2.Acker, A., ‘An isoperimetric inequality involving conformal mapping’, Proc. Amer. Math. Soc. 25 (1977), 230–234.CrossRefGoogle Scholar
3.Acker, A., ‘Some free boundary optimization problems and their solutions’, Numerische Behandlung von Differential-gleichungen mit besonderer Berücksichtigung freier Randwertaufgaben, Hg. Albrecht, von J., Collatz, L., Hämmerlin, G., pp. 9–22 (Birkhäuser Verlag, Basel, 1978).CrossRefGoogle Scholar
4.Beurling, A., ‘Free boundary problems for the Laplace equation’, Institute for Advanced Study Seminar, pp. 248–263 (Princeton, N.J., 1957).Google Scholar
5.Tepper, D. E., ‘Free boundary problem’, SIAM J. Math. Anal. 5 (1974), 841–846.CrossRefGoogle Scholar
6.Tepper, D. E., ‘On a free boundary problem, the starlike case’, SIAM J. Math. Anal. 6 (1975), 503–505.CrossRefGoogle Scholar
7.Tepper, D. E. and Wildenberg, G., ‘Some infinite free boundary problems’, Trans. Amer. Math. Soc. 248 (1979), 135–144.CrossRefGoogle Scholar