Let $\Omega \in \mathbb{R}^3$ be a bounded smooth domain. Assume that smooth functions $\sigma_1,\sigma_2$ satisfy $\sigma_1-\sigma_2 \in C_0^\infty(\Omega)$ and $\lambda\leq \sigma_1, \sigma_2 \leq \Lambda$, for some positive constants $\lambda,\Lambda$. Consider the solution $E_k, k=1,2$ of the following system: \begin{aligned} \nabla\cdot(\sigma_k E_k) = & 0, & x\in \Omega, \\ \nabla\times E_k = & B_0, & x\in \Omega, \\ \sigma_k E_k \cdot\nu = & 0 , & x\in \partial\Omega, \end{aligned} where $B_0 = (0,0,1)$ and the integral $$ I= \int_\Omega (\sigma_1 - \sigma_2) \nabla \cdot ((\sigma_1+\sigma_2)(E_1 - E_2) \times B_0) \, d x. $$ My question is: is the integral $I$ positive or does the estimate $|2I| \leq \|\sigma_1 -\sigma_2\|^2_{L^2(\Omega)}$ hold true.

Thanks.