Critical Sets of Solutions of Elliptic Equations in Periodic Homogenization

TL;DR

This paper investigates the structure of critical sets of solutions to elliptic equations with periodic coefficients, establishing uniform bounds on their Hausdorff measures and Minkowski contents through harmonic approximation and homogenization techniques.

Contribution

It provides the first uniform bounds on the Hausdorff measure of critical sets for solutions in periodic homogenization, using harmonic approximation and projection estimates.

Findings

Hausdorff measures of critical sets are uniformly bounded in periodic homogenization.

A new estimate of 'turning' for spherical harmonic projections is established.

Uniform bounds for Minkowski contents of critical sets are derived.

Abstract

In this paper we study critical sets of solutions $u_\e$ of second-order elliptic equations in divergence form with rapidly oscillating and periodic coefficients. We show that the $(d-2)$-dimensional Hausdorff measures of the critical sets are bounded uniformly with respect to the period $\e$, provided that doubling indices for solutions are bounded. The key step is an estimate of "turning " for the projection of a non-constant solution $u_\e$ onto the subspace of spherical harmonics of order $\ell$, when the doubling index for $u_\e$ on a sphere $\partial B(0, r)$ is trapped between $\ell -\delta$ and $\ell +\delta$, for $r$ between $1$ and a minimal radius $r^*\ge C_0\e$. This estimate is proved by using harmonic approximation successively. With a suitable $L^2$ renormalization as well as rescaling we are able to control the accumulated errors introduced by homogenization and projection. Our proof also gives uniform bounds for Minkowski contents of the critical sets.