= Solution
Take a <compactly supported> smooth variation $\varphi\in C_c^\infty(\Omega)$ and set $u_t=u+t\varphi$. Its support is away from the boundary, so it preserves any prescribed boundary values. Differentiating the <functional> at $t=0$ gives the <first variation>
$$
0=\left.\frac{d}{dt}\mathcal F(u_t)\right|_{t=0}=\int_\Omega\left[F_z(x,u,Du)\varphi+F_{p_i}(x,u,Du)D_i\varphi\right]dx.
$$
By <integration by parts>, there is no boundary term. The <fundamental lemma of the calculus of variations> therefore gives the <Euler-Lagrange equation>
$$
\boxed{D_i\bigl(F_{p_i}(x,u,Du)\bigr)=F_z(x,u,Du)\quad\text{in }\Omega.}
$$
Expanding the total derivative, an equivalent classical form is
$$
\boxed{F_{p_ip_j}(x,u,Du)D_{ij}u+F_{p_i z}(x,u,Du)D_i u+F_{p_i x_i}(x,u,Du)-F_z(x,u,Du)=0.}
$$
Repeated indices are summed. In the last term involving $x_i$, the derivative holds $z,p$ fixed; the other terms account for their dependence on $u(x),Du(x)$. <Compactly supported> variations derive the interior equation without imposing a natural boundary condition.
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