On an elliptic equation arising from composite materials

TL;DR

This paper establishes interior Schauder estimates for divergence form elliptic equations with piecewise H"older coefficients in two dimensions, showing solutions are piecewise smooth with bounded derivatives, thus resolving a question by Li and Vogelius.

Contribution

It provides the first interior Schauder estimates for elliptic equations with piecewise H"older coefficients in 2D, confirming piecewise smoothness of solutions.

Findings

Solutions are piecewise smooth with bounded derivatives when coefficients are piecewise constant.

The paper answers a previously open question in the theory of elliptic equations.

Provides a new interior regularity estimate for elliptic equations with discontinuous coefficients.

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 H\"older 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.