| Abstract: |
| Given a function $g$ in the homogeneous fractional Sobolev space $\dot H^{1/2, 2}(R,S^1)$ from the real line into the unit circle, by the direct minimization method, one can construct a half-harmonic map $u$ from the real line into the unit circle such that $u=g$ outside $(-1,1)$. In this talk we will show the existence of another half-harmonic maps with the same boundary condition $g$ by minimizing the fractional Dirichlet energy in a different homotopy class. We will also show that in general it is not possible to minimize the energy in every homotopy class. |
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