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

 

A bifurcation result for harmonic maps from an annulus to $S^2$ with not symmetric boundary data


Author: C. Greco
Journal: Proc. Amer. Math. Soc. 129 (2001), 1199-1206
MSC (2000): Primary 58E20
Published electronically: November 21, 2000
MathSciNet review: 1709752
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Abstract:

We consider the problem of minimizing the energy of the maps $u(r,\theta)$ from the annulus $\Omega_\rho=B_1\backslash\bar B_\rho$ to $S^2$ such that $u(r,\theta)$ is equal to $(\cos\theta,\sin\theta,0)$ for $r=\rho$, and to $(\cos(\theta+\theta_0)$, $\sin(\theta+\theta_0),0)$ for $r=1$, where $\theta_0\in[0,\pi]$ is a fixed angle.

We prove that the minimum is attained at a unique harmonic map $u_\rho$which is a planar map if $\log^2\rho+3\theta_0^2\le\pi^2$, while it is not planar in the case $\log^2\rho+\theta_0^2>\pi^2$.

Moreover, we show that $u_\rho$ tends to $\bar v$ as $\rho\to 0$, where $\bar v$ minimizes the energy of the maps $v(r,\theta)$ from $B_1$ to $S^2$, with the boundary condition $v(1,\theta)=(\cos(\theta+\theta_0)$, $\sin(\theta+\theta_0),0)$.


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Additional Information

C. Greco
Affiliation: Dipartimento di Matematica, Università degli Studi di Bari, Via Orabona 4, 70125 Bari, Italy
Email: greco@pascal.dm.uniba.it

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05643-4
PII: S 0002-9939(00)05643-4
Keywords: Harmonic maps, Dirichlet problem
Received by editor(s): October 16, 1998
Published electronically: November 21, 2000
Additional Notes: The author was supported in part by MURST and GNAFA of CNR
Communicated by: Linda Keen
Article copyright: © Copyright 2000 American Mathematical Society