Structure of singularities in the nonlinear nerve conduction problem

Abstract

We give a characterization of the singular points of the free boundary $\partial \{u>0\}$ for viscosity solutions of the nonlinear equation

$F(D^{2}u)=-{\chi }_{\{u>0\}},$

where $F$ is a fully nonlinear elliptic operator and $\chi$ is the characteristic function. This equation models the propagation of a nerve impulse along an axon. We analyze 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 Dipierro and the author’s 2018 paper. In particular, when $n=2$ we show that near a flat singular free boundary point, $\partial \{u>0\}$ is a union of four $C^1$ arcs tangential to a pair of crossing lines.