A bifurcation result for harmonic maps from an annulus to 𝑆² with not symmetric boundary data

Greco, C.

doi:10.1090/S0002-9939-00-05643-4

A bifurcation result for harmonic maps from an annulus to $S^2$ with not symmetric boundary data
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Proc. Amer. Math. Soc. 129 (2001), 1199-1206 Request permission

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 $v(1,\theta )=(\cos (\theta +\theta _0)$, $\sin (\theta +\theta _0),0)$.

References

F. Bethuel, H. Brezis, B. D. Coleman, and F. Hélein, Bifurcation analysis of minimizing harmonic maps describing the equilibrium of nematic phases between cylinders, Arch. Rational Mech. Anal. 118 (1992), no. 2, 149–168. MR 1158933, DOI 10.1007/BF00375093
Haïm Brezis and Jean-Michel Coron, Large solutions for harmonic maps in two dimensions, Comm. Math. Phys. 92 (1983), no. 2, 203–215. MR 728866, DOI 10.1007/BF01210846
Shmuel Kaniel and Itai Shafrir, A new symmetrization method for vector valued maps, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 4, 413–416 (English, with English and French summaries). MR 1179048
Étienne Sandier and Itai Shafrir, On the symmetry of minimizing harmonic maps in $N$ dimensions, Differential Integral Equations 6 (1993), no. 6, 1531–1541. MR 1235210
Etienne Sandier and Itai Shafrir, On the uniqueness of minimizing harmonic maps to a closed hemisphere, Calc. Var. Partial Differential Equations 2 (1994), no. 1, 113–122. MR 1384397, DOI 10.1007/BF01234318