Consider the linear elliptic operator of the form $$Lu=a^{ij}D_{ij}u+b^iD_iu$$ with coefficient satisfying $$\lambda I \leq [a^{ij}] \leq \lambda^{-1} I , \,\,\,\,\,\,\,\,\,\,\,\,\,\, |b^i|\leq \mu$$ where the inequalities means, for instance, that the matrix $[a^{ij}]-\lambda I$ is semipositive define, and $\lambda,\mu$ are positive constants.
Let $u$ satisfying $Lu+f\geq0.$
Let $Q(x)\in O(n)$ (i.e., orthogonal matrices), such that $$Q(x)^t(D^2u(x))Q(x)=diag(\lambda_1,\lambda_2,...,\lambda_k,-\lambda_{k+1},...,-\lambda_n)$$ where the $\lambda_i$ are positive and are the eigenvalues of the Hessian matrix $D^2u$.
Set $$a_*(x)=Q(x)^t diag(\frac{1}{2}\lambda,...,\frac{1}{2}\lambda,2\lambda^{-1},...2\lambda^{-1})Q(x).$$ Show the following inequality: $$\lambda/2|D^2u|\leq (a_*^{ij}-a^{ij})D_{ij}u$$