The rectified n-harmonic map flow with applications to homotopy classes

We introduce a rectified $n$-harmonic map flow from an n-dimensional closed Riemannian manifold to another closed Riemannian manifold. We prove existence of a global solution, which is regular except for a finite number of points, of the rectified n-harmonic map flow and establish an energy identity for the flow at each singular time. Finally, we present two applications of the rectified n-harmonic map flow to minimizing the n-energy functional and the Dirichlet energy functional in a homotopy class.Comment: 29 pages. A revised versio

Convergence of the Ginzburg-Landau approximation for the Ericksen-Leslie system

We establish the local well-posedness of the general Ericksen-Leslie system in liquid crystals with the initial velocity and director field in $H^1 \times H_b^2$. In particular, we prove that the solutions of the Ginzburg-Landau approximation system converge smoothly to the solution of the Ericksen-Leslie system for any $t \in (0,T^\ast)$ with a maximal existence time $T^\ast$ of the Ericksen- Leslie system