Nodal Sets and Doubling Conditions in Elliptic Homogenization

This paper is concerned with uniform measure estimates for nodal sets of solutions in elliptic homogenization. We consider a family of second-order elliptic operators {L-epsilon} in divergence form with rapidly oscillating and periodic coefficients. We show that the (d-1)-dimensional Hausdorff measures of the nodal sets of solutions to L-epsilon(u(epsilon)) = 0 in a ball in (d) are bounded uniformly in epsilon > 0. The proof relies on a uniform doubling condition and approximation of u(epsilon) by solutions of the homogenized equation.