# Structure of singularities in the nonlinear nerve conduction problem

## TL;DR

This paper characterizes the structure of singular points on the free boundary in a nonlinear nerve conduction model, revealing geometric properties and stratification near these points, especially in two dimensions.

## Contribution

It provides a detailed analysis of the free boundary's singularities in a nonlinear PDE model of nerve impulses, including geometric descriptions and stratification methods.

## Key findings

- Near flat singular points, the free boundary consists of four $C^1$ arcs in 2D.
- In the linear case, the singular set decomposes into degenerate and flat points.
- Blow-ups of solutions at certain points are homogeneous functions.

## Abstract

We give a characterisation of the singular points of the free boundary $\partial \{u>0\}$ for viscosity solutions of the nonlinear equation \begin{equation}F(D^2 u)=-\chi_{\{u>0\}},\tag{0.1} \end{equation} where $F$ is a fully nonlinear elliptic operator and $\chi$ the characteristic function. The equation (0.1) models the propagation of a nerve impulse along an axon. We analyse the structure of the free boundary $\partial\{ u>0\}$ near the singular points where $u$ and $\nabla u$ vanish simultaneously. Our method uses the stratification approach developed in [DK18]. In particular, when $n=2$ we show that near a rank-2 flat singular free boundary point $\partial\{ u>0\}$ is a union of four $C^1$ arcs tangential to a pair of crossing lines. Moreover, if $F$ is linear then the singular set of $\partial\{ u>0\}$ is the union of degenerate and rank-2 flat points. We also provide an application of the boundary Harnack principles to study the higher order flat degenerate points and show that if $\{u<0\}$ is a cone then the blow-ups of $u$ are homogeneous functions.