Critical sets of elliptic equations

TL;DR

This paper develops new quantitative techniques to estimate the size and structure of the critical set of solutions to elliptic equations, improving understanding even for harmonic functions.

Contribution

It introduces a quantitative stratification method for analyzing the critical set of elliptic solutions, providing effective volume and measure estimates.

Findings

Effective volume estimates for tubular neighborhoods of stratified critical sets

Refined Hausdorff measure bounds under regularity assumptions

Applicable to harmonic functions and general 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 $u$.