Structure of singularities in the nonlinear nerve conduction problem

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.