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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Dirichlet finite biharmonic functions on the unit disk with distorted metrics


Author: Heppé O’Malla
Journal: Proc. Amer. Math. Soc. 32 (1972), 521-524
MSC: Primary 31A30
DOI: https://doi.org/10.1090/S0002-9939-1972-0340627-1
MathSciNet review: 0340627
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Abstract: The Riemannian manifold $ {D_\alpha }$ obtained from the unit disk D by giving the distorted metric $ {(1 - \vert z\vert)^{ - \alpha }}\vert dz\vert$ does not admit any Dirichlet finite nonharmonic biharmonic function if and only if $ \alpha \geqq \tfrac{3}{4}$.


References [Enhancements On Off] (What's this?)

  • [1] M. Nakai, The equation $ \Delta u = Pu$ on the unit disk with almost rotation free $ P \geqq 0$, J. Differential Equations 10 (1972). MR 0296285 (45:5346)
  • [2] M. Nakai and L. Sario, Existence of Dirichlet finite biharmonic functions (to appear). MR 0425831 (54:13781)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0340627-1
Keywords: Dirichlet integral, biharmonic, harmonic, Riemannian manifold, classification theory of Riemann surfaces
Article copyright: © Copyright 1972 American Mathematical Society

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