Euler–Lagrange for a double integral: one divergence term per variable

Consider the functional $I(u(x,y))=\iint_GF(x,y,u,u_x,u_y)\,dx\,dy$ over the region of integration G, where $u$ is continuous and has continuous derivatives upto second order. Further $u$ takes prescribed values on the boundary of G. Then the differential equation for the extremization of the functional is: