Electron beams: partially flat solutions of a nonlinear elliptic equation with a singular absorption term

TL;DR

This paper rigorously analyzes partially flat solutions of a nonlinear elliptic equation with a singular absorption term, extending previous studies on electron beam models with edge effects and singular current distributions.

Contribution

It provides a rigorous mathematical framework for understanding partially flat solutions and the role of singularities in the current distribution near the cathode edge.

Findings

Existence of partially flat solutions under certain conditions.

Necessity of singularity in current near the cathode edge.

Extension of previous models to include edge effects and singular behaviors.

Abstract

In the so-called Child-Langmuir law, established since 1911, an electron beam is formed linking two electrodes, which are assumed to be two parallel plates of area $A$, separated to a finite distance $D.$ When $% D\ll \sqrt{A},$ \textquotedblleft edge effects\textquotedblright\ are negligible and the modelling is reduced to a nonlinear boundary problem for a singular ordinary differential equation\ in which a constant coefficient (the generated electric current $j$) must be found in order to get simultaneously Dirichlet and Neumann homogeneous boundary conditions in one of the extremes. If $D>\sqrt{A},$ then the problem becomes much more difficult since the \textquotedblleft edge effects\textquotedblright\ arise in the plane $(x,y)$ and the electric current (now $j(x)$ due to the presence of a very large perpendicular magnetic field) must be determined in order to get solutions $u(x,y)$ of a singular semilinear equation which are partially flat ($u=\frac{\partial u}{\partial n}=0$ on a part of the boundary). In this paper, we offer a rigorous mathematical treatment of some former studies (Joel Lebowitz and Alexander Rokhenko (2003) and Alexander Rokhenko (2006)), where several open questions were left open: for instance, the need for a singularity of $j(x)$ near the cathode edge to get such partially flat solutions.