Uniqueness for the harmonic map flow from surfaces to general targets
Given a compact surface \(M\) with boundary \(\partial M\) and a compact Riemannian manifold \(N\) embedded in some space \(\mathbb{R}^k\), the harmonic map heat flow is the problem of finding a (weak) solution \(u: M\times (0, T)\to N\) of \[ \begin{cases} {\partial \over \partial t} u- \Delta_M u= A(u) (\nabla u,\nabla u) &\text{on } M\times (0,T),\\ u(x,t)= \gamma (x) &\text{for }t\geq 0,\;x\in \partial M,\\ u(x, 0)= u_0, &x\in M, \end{cases} \tag{1} \] with given initial and boundary data. The author shows: if \(u\) and \(v\) are weak solutions of (1) in \(H^1 (M\times (0,T), N)\) whose energy is non-increasing in time then \(u=v\).
scientific article; zbMATH DE number 790584