Dirichlet finite biharmonic functions on the unit disk with distorted metrics

O’Malla, Heppé

doi:10.1090/S0002-9939-1972-0340627-1

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Dirichlet finite biharmonic functions on the unit disk with distorted metrics
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Proc. Amer. Math. Soc. 32 (1972), 521-524 Request permission

Abstract:

The Riemannian manifold ${D_\alpha }$ obtained from the unit disk D by giving the distorted metric ${(1 - |z|)^{ - \alpha }}|dz|$ does not admit any Dirichlet finite nonharmonic biharmonic function if and only if $\alpha \geqq \tfrac {3}{4}$.

References

Mitsuru Nakai, The equation $\Delta u=Pu$ on the unit disk with almost rotation free $P\geq 0$, J. Differential Equations 11 (1972), 307–320. MR 296285, DOI 10.1016/0022-0396(72)90047-2
Mitsuru Nakai and Leo Sario, Existence of Dirichlet finite biharmonic functions, Ann. Acad. Sci. Fenn. Ser. A. I. 532 (1973), 34. MR 425831