Let $A$ be a symmetric, uniformly elliptic constant matrix with $\lambda \leq A \leq \Lambda$. Consider the weak solution (which is smooth from standard elliptic theory) $u \in W_{loc}^{1,2}$ solving $\text{div} A \nabla u = 0$ and let $S_1$ be the unit sphere in $\mathbb{R}^n$. I am trying to show the following inequality: $$ \int_{S_1} \langle A\nabla u, x\rangle \langle \Lambda \nabla u - A \nabla u, x\rangle d\sigma \geq 0. $$
The few examples I found, seemed to satisfy this inequality, and I want to know if this is true in general? Clearly, when $A$ is the identity matrix, this is trivially true since $\Lambda =1$ and $A = \mathbf{I}$ which gives $\langle \Lambda \nabla u - A \nabla u, x\rangle = \langle \nabla u - \nabla u, x\rangle =0$.
