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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.

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  • $\begingroup$ how did you come up with this particular quantity? $\endgroup$
    – username
    Commented Jun 29, 2014 at 8:45
  • $\begingroup$ I encountered this problem while trying to solve an inverse problem involving the above system. $I$ is a remainder term in the proof of uniqueness. $I$ is also of the form $$I= -\int_\Omega (\sigma_1 + \sigma_2) \nabla \cdot ((\sigma_1-\sigma_2)(E_1 - E_2) \times B_0) \, d x $$ $\endgroup$
    – Lingyun
    Commented Jun 29, 2014 at 13:42
  • $\begingroup$ you have a very precise idea of what the estimate should be: can you detail the steps you did/ could not do? $\endgroup$
    – username
    Commented Jul 2, 2014 at 21:23
  • $\begingroup$ First, I guess that the $2I$ should have an upper bound $\|\sigma_1 - \sigma_2\|^2$. Then, I notice, from some numerical examples, the positiveness of $I$. I tried to prove that $(\sigma_1 + \sigma_2) \nabla \cdot((\sigma_1 - \sigma_2)(E_1 -E_2)\times B_0)$ is non-positive everywhere, but the fact that integral of $\nabla \cdot((\sigma_1 - \sigma_2)(E_1 -E_2)\times B_0)$ over $\Omega$ is zero implies it changes sign in $\Omega$. Now, I am trying to using the PDE to construct some flows such that 1.) these flows covers $\Omega$; 2.) integral over any flow is positive. $\endgroup$
    – Lingyun
    Commented Jul 20, 2014 at 14:08

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