# 稳定态、极小曲面与拉普拉斯方程

1，稳定态，就是物体的位置或者性质与时间无关。与时间无关，则关于时间的导数为 $$0$$，$$u_t=0, u_{tt}=0$$。

2，极小曲面：我们也可以通过简化极小曲面方程来得到拉普拉斯方程。

$J(\epsilon)=J(u+\epsilon v)=\iint_{\Omega} \sqrt{(u_x+\epsilon v_x)^2+(u_y+\epsilon v_y)^2+1}dA$

\begin{align*}J'(\epsilon)&=\iint_{\Omega}\frac{(u_x+\epsilon v_x)v_x+(u_y+\epsilon v_y)v_y}{\sqrt{(u_x+\epsilon v_x)^2+ (u_y+\epsilon v_y)^2 +1}}dA\end{align*}

\begin{align*}J'(0)&=\iint_{\Omega}\frac{u_xv_x+u_yv_y}{\sqrt{u_x^2+ u_y^2 +1}}dA\\ &=\iint_{\Omega}\frac{(u_x+\epsilon v_x)v_x+(u_y+\epsilon v_y)v_y}{\sqrt{(u_x+\epsilon v_x)^2+ (u_y+\epsilon v_y)^2 +1}}dA\\ &=\iint_{\Omega}\frac{1} {\sqrt{u_x^2+ u_y^2 +1}} \nabla u\cdot \nabla vdA\end{align*}

\begin{align*}J'(0)&=\oint_{\partial\Omega}\frac{v}{\sqrt{ u_x^2+ u_y^2 +1 }}\cdot\frac{\partial u}{\partial \vec{n}}dA-\iint_{\Omega}v\cdot\left[\frac{\partial}{\partial x}\left(\frac{u_x}{ \sqrt{u_x^2+ u_y^2 +1} }\right)+ \frac{\partial}{\partial y}\left(\frac{u_y}{ \sqrt{u_x^2+ u_y^2 +1} }\right) \right]dA\end{align*}

$\frac{\partial}{\partial x}\left(\frac{u_x}{ \sqrt{u_x^2+ u_y^2 +1} }\right)+ \frac{\partial}{\partial y}\left(\frac{u_y}{ \sqrt{u_x^2+ u_y^2 +1} }\right) =0$

\begin{align*}J^{\prime\prime}(\epsilon)=\iint_{\Omega}\frac{v_x^2+v_y^2+[v_x(u_y+\epsilon v_y)-v_y(u_x+\epsilon v_x)]^2}{((u_x+\epsilon v_x)^2+(u_y+\epsilon v_y)^2+1)^{3/2}}\ge 0\end{align*}

$\frac{\partial}{\partial x}\left(\frac{u_x}{ \sqrt{u_x^2+ u_y^2 +1} }\right)+ \frac{\partial}{\partial y}\left(\frac{u_y}{ \sqrt{u_x^2+ u_y^2 +1} }\right) =0$

$u_{xx}+u_{yy}=0$