= Solution
Let $u\mapsto u+\varepsilon\eta$ and $v\mapsto v+\varepsilon\xi$, where the variations vanish on the boundary. The <first variation> is
$$
\begin{aligned}
\delta\mathcal L
=\iint_\Omega \bigl(&f_u\eta+f_v\xi
+f_{u_x}\eta_x+f_{u_y}\eta_y\\
&+f_{v_x}\xi_x+f_{v_y}\xi_y)\,dx\,dy.
\end{aligned}
$$
Applying <integration by parts> to the four derivative terms and discarding the boundary contributions gives
$$
\delta\mathcal L=\iint_\Omega
\left(f_u-\frac{\partial f_{u_x}}{\partial x}
-\frac{\partial f_{u_y}}{\partial y}\right)\eta\,dx\,dy
$$
$$
{}+\iint_\Omega
\left(f_v-\frac{\partial f_{v_x}}{\partial x}
-\frac{\partial f_{v_y}}{\partial y}\right)\xi\,dx\,dy.
$$
The variations $\eta$ and $\xi$ are independent and arbitrary in the interior. The <fundamental lemma of the calculus of variations> therefore gives the two <Euler-Lagrange equations for two fields>
$$
\boxed{f_u-\partial_x f_{u_x}-\partial_y f_{u_y}=0},
$$
$$
\boxed{f_v-\partial_x f_{v_x}-\partial_y f_{v_y}=0}.
$$
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