Pointwise estimates on the gradients in 2D media containing smooth inclusions : An integral equation approach
Résumé
We consider a composite medium that contains a pair of smooth inclusions separated by a distance $\delta>0$. We revisit the regularity results of Li-Vogelius and Li-Nirenberg from the point of view of integral equations for the potential densities. We show that when the inclusions are $C^{1,\alpha}$, the system is invertible in $C^{0,\alpha'}$ uniformly in $\delta$ , for any $\alpha'<\alpha$. In the particular case when the inclusions are disks, we characterize the spectrum of the system of integral equations and relate its behavior as $\delta\to0$ to the bounds on the potential.