An introduction to the study of critical points of solutions of elliptic and parabolic equations

TL;DR

This paper provides an introductory survey of classical and recent results on the critical points of solutions to elliptic and parabolic PDEs, focusing on boundary value problems like Laplace's and heat equations.

Contribution

It offers a simplified overview of key findings related to the size, number, and location of critical points in solutions of fundamental PDEs, highlighting both historical and recent developments.

Findings

Estimation of the local size of the critical set

Dependence of critical points on boundary values and domain geometry

Location of critical points within the domain

Abstract

We give a survey at an introductory level of old and recent results in the study of critical points of solutions of elliptic and parabolic partial differential equations. To keep the presentation simple, we mainly consider four exemplary boundary value problems: the Dirichlet problem for the Laplace's equation; the torsional creep problem; the case of Dirichlet eigenfunctions for the Laplace's equation; the initial-boundary value problem for the heat equation. We shall mostly address three issues: the estimation of the local size of the critical set; the dependence of the number of critical points on the boundary values and the geometry of the domain; the location of critical points in the domain.