Let $D$ be a bounded region of $\mathbb{R}^{2}$, with boundary $\partial D$. Let $u(x, y)$ be a smooth function defined on $D$, subject to the boundary condition that $u=0$ on $\partial D$ and the normalization condition that

$$\int_{D} u^{2} d x d y=1$$

Let $I[u]$ be the functional

$$I[u]=\int_{D}|\nabla u|^{2} d x d y$$

Show that $I[u]$ has a stationary value, subject to the stated boundary and normalization conditions, when $u$ satisfies a partial differential equation of the form

$$\nabla^{2} u+\lambda u=0$$

in $D$, where $\lambda$ is a constant.

Determine how $\lambda$ is related to the stationary value of the functional $I[u]$. $[$ Hint: Consider $\boldsymbol{\nabla} \cdot(u \boldsymbol{\nabla} u)$.]