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

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

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

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

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

where $k$ is a constant and $u$ is prescribed on $\partial \mathcal{D}$.