Porblem: Let $\Omega\in \mathbb R^n$ be a domain and $F(x,p)\in C^3(\Omega\times \mathbb R^n)$ be locally uniformly convex in the $p$ variables. Suppose that $u$, $v\in C^2(\Omega)$ be critical points of the functional $$ \mathcal F[u]:=\int_\Omega F(x, Du)\, dx. $$ Prove that if $u-v$ attains a local maximum or a local minimum, then it is a constant.
My attempt: I have shown that the Euler-Lagrange equation for this problem is $$ \nabla_x\cdot (\nabla_p F(x,Du))=\sum_{i,j} (\partial_{p_i}\partial_{p_j} F )(\partial_{x_i}\partial_{x_j} u)+\sum_i\partial_{x_i}\partial_{p_i}F=0. $$ At the local extrema, we know that $F(x,Du)=F(x,Dv)$ so I managed to show that $D^2(u-v)=0$ from the local uniform convexity of $F$. But this is not enough to prove the result.
Any help is appreciated.