Critical sets of elliptic equations

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 $\text{Cr}(u)\equiv \{x:\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }(x)=0\}$. The results are new even for harmonic functions on ${\text{dR}}^n$. Given such a $u$, the standard {\it first order} stratification $\{{\text{cS}}^k\}$ 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 $\{{\text{cS}}_{\eta ,r}^k\}$ 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 ${\text{cS}}_{\eta ,r}^k$, which lead directly to $(n-2+ϵ)$-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 $u$.