I want to calculate the $\frac{d\lambda}{dt}$ under volume preserving mean curvature flow. $\lambda$ is first eigenvalue of Laplacain i.e. \begin{align} -\Delta u &=\lambda(t) u ~~~~~~~~~~~in ~~\Omega_t ~~~~~~~~~~~~~~&(1) \\ u|_{\partial\Omega_t}&=0 &(2) \\ \int_{\Omega_t} u^2 &=1 &(3) \end{align} $\Delta$ is Euclidian Laplacian. $\Omega_t$ is the domain enclosed by $M_t$. The notation can be found in picture below or in Huisken, Gerhard, The volume preserving mean curvature flow, J. Reine Angew. Math. 382, 35-48 (1987). ZBL0621.53007.

I try two ways to calculate $\frac{d\lambda}{dt}$, but I get different results.And fail to find my mistake.

First way, differential $(1)$, I have $$ -\Delta(\partial_t u)=\partial_t \lambda u + \lambda \partial_t u $$ multiply $u$ in two side and integral $$ -\int_{\Omega_t} u \Delta(\partial_t u) =\partial_t \lambda +\int_{\Omega_t} \lambda u\partial_t u $$ although $\Omega_t$ is changing, but from $(3)$, I have $$ \int_{\Omega_t} u\partial_t u =0 $$ Then, \begin{align} \int_{\Omega_t} u\Delta(\partial_t u ) &=-\int_{\Omega_t} \nabla u\cdot \nabla(\partial_t u) \\ &=-\int_{\partial \Omega_t}\frac{\partial u}{\partial\nu} \partial_t u + \int_{\Omega_t} \Delta u \partial_t u \\ &=-\int_{\partial \Omega_t}\frac{\partial u}{\partial\nu} \partial_t u \end{align} so, I have $$ \partial_t\lambda =\int_{\partial \Omega_t}\frac{\partial u}{\partial\nu} \partial_t u $$

Second way, we know $$ \lambda=\int_{\Omega_t} |\nabla u|^2=\int_S\int_0^{|F(x,t)|} |\nabla u|^2\rho^n d\rho dx $$ $S$ is n-dim sphere, and I asuume origin is in $\Omega_t$. Then \begin{align} \partial_t \lambda &= \int_S |\nabla u |^2 |F|^n \partial_t|F| dx+\int_S\int_0^{|F(x,t)|} 2\nabla u\nabla(\partial_tu)\rho^n d\rho dx \\ &=\int_{\partial\Omega_t}|\nabla u|^2 \partial_t|F| + 2\int_{\Omega_t} \nabla u\nabla u_t \\ &=\int_{\partial\Omega_t}|\nabla u|^2 (h-H)\frac{F\cdot \nu}{|F|} +2\int_{\partial\Omega_t} u_t \frac{\partial u}{\partial \nu} -2\int_{\Omega_t }\Delta u u_t \\ &=\int_{\partial\Omega_t}|\nabla u|^2 (h-H)\frac{F\cdot \nu}{|F|} +2\int_{\partial\Omega_t} u_t \frac{\partial u}{\partial \nu} \end{align}

I feel they are not equal, so at least one is wrong. Where is it? Besides, this question is a little long, if there are any thing which is unclear, please ask me . Thanks.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

enter image description here


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.