Critical Sets of Elliptic Equations with Rapidly Oscillating Coefficients in Two Dimensions

TL;DR

This paper investigates the structure and measure of critical sets of solutions to elliptic equations with rapidly oscillating periodic coefficients in two dimensions, using a new approach based on reducing doubling indices.

Contribution

It introduces a novel method leveraging the properties of harmonic polynomials in two dimensions to analyze critical sets, differing from previous tangent plane control techniques.

Findings

Critical sets have bounded Hausdorff measure independent of oscillation period

Critical points of harmonic polynomials in 2D contain only one point

New approach simplifies analysis of oscillating elliptic equations in 2D

Abstract

In this paper we continue the study of critical sets of solutions $u_\e$ of second-order elliptic equations in divergence form with rapidly oscillating and periodic coefficients. In \cite{Lin-Shen-3d}, by controling the "turning" of approximate tangent planes, 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. In this paper we use a different approach, based on the reduction of the doubling indices of $u_\e$, to study the two-dimensional case. The proof relies on the fact that the critical set of a homogeneous harmonic polynomial of degree two or higher in dimension two contains only one point.