Axially symmetric harmonic maps and relaxed energy
Hardt, Robert M.
Doctor of Philosophy
Here we investigate some new phenomena in harmonic maps that result by imposing a symmetry condition. A map $u:B\sp3\to S\sp2$ is called axially symmetric if, in cylindrical coordinates, $u(r,\theta,z)$ = $(\cos\theta\sin\phi,\sin\theta\sin\phi,\cos\phi)$ for some real-valued function $\phi(r,z)$, called an angle function for u. The important notion of the L energy of a map from $B\sp3$ to $S\sp2$ was first studied by H. Brezis, F. Bethuel, and J. M. Coron. In (BBC), the weak $H\sp1$ lower semicontinuity of $\rm E + 8\pi\lambda L$ is proven. Thus, the minimizers of $\rm E + 8\pi\lambda L$ exist. For minimizers of $\rm E + 8\pi\lambda L$, 0 $<$ $\lambda$ $<$ 1, Bethuel and Brezis (BB) prove that the singularities are only isolated points. Note that such minimizers are still weak solutions of the harmonic map equation. In this thesis, we treat these problems in the axially symmetric context. By studying a elliptic equation, we show that there is at most one smooth axially symmetric harmonic map corresponding to any given smooth axially symmetric boundary data. We also show that any minimizer in the axially symmetric class of $\rm E + 8\pi\lambda L$, where 0 $<$ $\lambda$ $\leq$ 1, has only isolate singularities in minimizers may occur even for $\lambda$ = 1. These provide the first examples of isolated singularities of degree 0.