English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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 $F$ isa fully nonlinear elliptic operator and $\chi$ the characteristic function. Theequation (0.1) models the propagation of a nerve impulse along an axon.  We analyse the structure of the free boundary $\partial\{ u>0\}$ near thesingular points where $u$ and $\nabla u$ vanish simultaneously. Our method usesthe stratification approach developed in [DK18].  In particular, when $n=2$ we show that near a rank-2 flat singular freeboundary point $\partial\{ u>0\}$ is a union of four $C^1$ arcs tangential to apair 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 studythe higher order flat degenerate points and show that if $\{u<0\}$ is a conethen the blow-ups of $u$ are homogeneous functions.

Structure of singularities in the nonlinear nerve conduction problem