Computational Methods and Function Theory 8 (2008), No. 1, 143--150
Copyright Heldermann Verlag 2008
Dmitry Khavinson
dkhavins@cas.usf.edu , University of South Florida, Department of Mathematics & Statistics, 4202 East Fowler Avenue, PHY114, Tampa, FL 33620-5700, U.S.A.
Harold S. Shapiro
shapiro@math.kth.se , Royal Institute of Technology, Department of Mathematics, Stockholm, S-10044, Sweden.
We investigate the problem of uniqueness for functions u harmonic in a domain Ω and vanishing on some parts of the intersection (not necessarily connected) of Ω with a line m. It turns out that for some configurations u must vanish on the whole intersection of m and Ω, but this is not always the case. Generalizations to solutions of more general analytic elliptic equations are discussed as well.
Keywords: Analytic continuation, harmonic functions, cells of harmonicity, Schwarz reflection principle.
MSC 2000: Primary 31A35, 35A20; Secondary 31A05.
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