A Morse Lemma for degenerate critical points of solutions of nonlinear equations in $\R^2$

TL;DR

This paper establishes a Morse Lemma for degenerate critical points of solutions to nonlinear elliptic equations in two dimensions, analyzing their nondegeneracy and level set geometry.

Contribution

It extends Morse theory to degenerate critical points of nonlinear PDE solutions in R^2, providing new insights into their structure.

Findings

Proves a Morse Lemma for degenerate critical points in R^2

Analyzes nondegeneracy conditions of critical points

Describes the shape of level sets near critical points

Abstract

In this paper we prove a Morse Lemma for degenerate critical points of a function u which satisfies -\Delta u=f(u) in B_1, where B_1 is the unit ball of R^2 and f is a smooth nonlinearity. Other results on the nondegeneracy of the critical points and the shape of the level sets are proved.