Derive the Euler-Lagrange equation for the function $u(x, y)$ which gives a stationary value of

$$I=\int_{\mathcal{D}} L\left(x, y, u, \frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}\right) d x d y$$

where $\mathcal{D}$ is a bounded domain in the $(x, y)$ plane, with $u$ fixed on the boundary $\partial \mathcal{D}$.

Find the equation satisfied by the function $u$ which gives a stationary value of

$$I=\int_{\mathcal{D}}\left[\left(\frac{\partial u}{\partial x}\right)^{2}+\left(\frac{\partial u}{\partial y}\right)^{2}\right] d x d y$$

with $u$ given on $\partial \mathcal{D}$.