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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Stiffness of harmonic functions


Author: Cecilia Y. Wang
Journal: Proc. Amer. Math. Soc. 77 (1979), 103-106
MSC: Primary 30F15
MathSciNet review: 539639
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Abstract: Harmonic functions cannot change rapidly. For example, if K is a compact subset of a Riemann surface R and {u} a family of harmonic functions u on R of nonconstant sign on K, then it is known that there exists a constant $ q \in (0,1)$ independent of u such that $ {\max _K}\vert u\vert \leqslant q{\sup _R}\vert u\vert$ for all $ u \in \{ u\} $. In the present note we shall show that relations expressing such ``stiffness'' of harmonic functions can also be given for the Dirichlet norm and for the partial derivative with respect to the Green's function.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1979-0539639-0
PII: S 0002-9939(1979)0539639-0
Article copyright: © Copyright 1979 American Mathematical Society