Critical sets of elliptic equations

Abstract

Given a solution $u$ to a linear homogeneous second order elliptic equation with Lipschitz coefficients, we introduce techniques for giving improved estimates of the critical set $\Cr(u)\equiv \{x:|\nabla u|(x)=0\}$. The results are new even for harmonic functions on $\dR^n$. Given such a $u$, the standard {\it first order} stratification $\{\cS^k\}$ of $u$ separates points $x$ based on the degrees of symmetry of the leading order polynomial of $u-u(x)$. In this paper we give a quantitative stratification $\{\cS^k_{\eta,r}\}$ of $u$, which separates points based on the number of {\it almost} symmetries of {\it approximate} leading order polynomials of $u$ at various scales. We prove effective estimates on the volume of the tubular neighborhood of each $\cS^k_{\eta,r}$, which lead directly to $(n-2+\epsilon)$-Minkowski content estimates for the critical set of $u$. With some additional regularity assumptions on the coefficients of the equation, we refine the estimate to a uniform $(n-2)$-Hausdorff measure estimate on the critical set of $u$.