Prove that $u=v$

Question

I got the following integral identity $$\int_{\Omega}\left[H(\nabla u)(\nabla H)(\nabla u)-H(\nabla v)(\nabla H)(\nabla v)\right]\cdot\nabla\left(u-v\right)\;dx=0$$ and i want to prove that $u=v$.

Note that $H$ is a Finsler norm, who is homogeneous of degree 1 and convex. How can I use that? Also, since every two norms are equivalent on $\mathbb{R}^N$ there exists a,b>0 so that $a|x|\leq H(x)\leq b|x|$.

Answer 1

Hint:

What is the derivative of the (nonlinear) functional $$ u \mapsto \frac 12 \, \int_\Omega H(\nabla u)^2 \, \mathrm{d}x $$ ?