An Elliptic Regularity Result for a Composite Medium with "Touching" Fibers of Circular Cross-Section

In this paper we consider the elliptic equation $\nabla \cdot a\nabla u= 0$ in a two dimensional domain $\Omega$, which contains a finite number of circular inhomogeneities (cross-sections of fibers). The coefficient, a, takes two constant values, one in all the inhomogeneities and one in the part of $\Omega$ which lies outside the inhomogeneities. A number of the inhomogeneities may possibly touch, but in spite of this we prove that any variational solution u (with sufficiently smooth boundary data) is in $W^{1,\infty}$. For this very interesting, particular type of coefficient, our result improves a classical regularity result due to DeGiorgi and Nash, which asserts that the solution is in the Hölder class $C^\gamma$ for some positive exponent $\gamma$.