An elliptic regularity result for a composite medium with "touching" fibers of circular cross-section

In this paper we consider the elliptic equation ▽ · α▽u = 0 in a two dimensional domain Ω, 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 Ω 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,∞. 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^γ for some positive exponent γ.