The nodal set of a function u is the set of points where u vanishes, while the critical set consists of points where the gradient of u vanishes. For a harmonic function or a solution of a second-order elliptic equation, it is known that the nodal set is of Hausdorff dimension (d-1) and the critical set is of dimension (d-2). Moreover, the Hausdorff measures of these sets depend on the doubling constants of the function.
In this talk I will describe my recent work, joint with Fanghua Lin, on geometric properties of solutions of partial differential equations in the homogenization theory. We consider second-order elliptic equations with rapidly oscillating and periodic coefficients. We show that the (d-1)-dimensional Hausdorff measures of the nodal sets and the (d-2)-dimensional Hausdorff measures of the critical sets are bounded uniformly with respect to the period, provided that the doubling constants for the solutions are bounded. The proof uses the harmonic approximation successively. The key is to control accumulated errors by renormalization and rescaling.