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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.

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