On an elliptic equation arising from composite materials

Abstract

In this paper, we derive an interior Schauder estimate for the divergence form elliptic equation \begin{equation*} D_i(a(x)D_iu)=D_if_i \end{equation*} in ${\mathbb{R}}^2$, where $a(x)$ and $f_i(x)$ are piecewise Hölder continuous in a domain containing two touching balls as subdomains. When $f_i\equiv 0$ and $a$ is piecewise constant, we prove that $u$ is piecewise smooth with bounded derivatives. This completely answers a question raised by Li and Vogelius [7] in dimension 2.

36 pages, 2 figures, submitted

References in corpus (2)

• Gradient estimates for parabolic and elliptic systems from linear laminates
• Gradient estimates for solutions of the Lamé system with partially infinite coefficients

Cited by in corpus (1)

• Gradient estimates for the insulated conductivity problem: the case of $m$-convex inclusions