Tag: alpha-调和映照

我们首先来看$\alpha$调和映照流, 其方程可以写作 $$\left\{\begin{aligned} \partial_t u^\beta-\Delta u^\beta-(\alpha-1) \frac{u_{kl}^{\beta\gamma}u_k^\gamma u_l^\beta}{1+\lvert \nabla u \rvert^2}&=\Gamma^\beta(u)(\nabla u,\nabla u),\quad x\in M;\\ u(\cdot,0)&=u_0\in C^\infty(M), \quad x\in M;\\ u^n(x,t)&=0, \quad (x,t)\in\partial M\times[0,T];\\ \frac{\partial u^\beta}{\partial\nu}&=0,\quad (x,t)\in \partial M\times [0,T],\quad \beta=1,2,\ldots, n-1. \end{aligned}\right.$$ 我们将考虑其线性形式, 即 $$ \partial_t u^\beta-\Delta u^\beta-(\alpha-1) \frac{u_{kl}^{\beta\gamma}w_k^\gamma w_l^\beta}{1+\lvert \nabla w \rvert^2}=\Gamma^\beta(w)(\nabla w,\nabla w),\quad x\in M. $$ 改写成标准的形式 $$ \mathcal{L}u=f, $$