Upper bound of critical sets of solutions of elliptic equations in the plane

TL;DR

This paper establishes upper bounds on the number of singular and critical points of solutions to elliptic equations in the plane, using Carleman estimates adapted from Donnelly and Fefferman.

Contribution

It provides new upper bounds for the measure of singular and critical sets of elliptic solutions in two dimensions, extending previous methods with polynomial-based Carleman estimates.

Findings

Finite number of singular points and critical points in solutions

Derived explicit upper bounds for these points

Extended Carleman estimate techniques to elliptic solutions

Abstract

In this note, we investigate the measure of singular sets and critical sets of real-valued solutions of elliptic equations in two dimensions. These singular sets and critical sets are finitely many points in the plane. Adapting the Carleman estimates involving polynomial functions at singularities by Donnelly and Fefferman in \cite{DF90}, we obtain the upper bounds of singular points and critical points.