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

Jesús Ildefonso Díaz

TL;DR

The paper analyzes 2D space-charge-limited electron beams between parallel plates with edge effects, formulating a singular semilinear elliptic problem $- abla^{2}u=\frac{j(x)}{\sqrt{u}}$ on a strip and imposing partially flat boundary behavior to reflect vanishing flux on part of the cathode. Using a sophisticated sub-/super-solution strategy and domain decomposition, it constructs a weak, nondegenerate solution under a near-edge singular current density $j(x)\sim\frac{A}{(-x)^{\beta}}$ with $0<\beta<\tfrac{1}{2}$ and $j(x)=0$ on the complementary side; the solution satisfies sharp near-boundary estimates $u(x,y)\gtrsim \delta^{4/3}$ and $u(0,y)\gtrsim y^{\alpha}$ with $\alpha=\tfrac{2}{3}(2-\beta)$. A key part is the analysis of an auxiliary nonlinear eigenvalue problem $-U''+\frac{V_0}{\sqrt{U}}=\lambda U$ on a finite interval, which exhibits a bifurcation from infinity and a non-smooth flat solution at a critical value, guiding the construction of subsolutions and the existence of a partially flat supersolution. The work proves existence of a global weak solution, nondegeneracy, and, via a parabolic comparison framework, uniqueness in the nondegenerate class, thereby rigorously justifying the formation of current wings near cathode edges and extending the one-dimensional Child-Langmuir theory to 2D with edge effects. These results provide a rigorous foundation for edge phenomena in space-charge-limited devices and offer a framework for further analysis of 2D electrostatic-flow problems with singular absorptions.

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.

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

TL;DR

The paper analyzes 2D space-charge-limited electron beams between parallel plates with edge effects, formulating a singular semilinear elliptic problem on a strip and imposing partially flat boundary behavior to reflect vanishing flux on part of the cathode. Using a sophisticated sub-/super-solution strategy and domain decomposition, it constructs a weak, nondegenerate solution under a near-edge singular current density with and on the complementary side; the solution satisfies sharp near-boundary estimates and with . A key part is the analysis of an auxiliary nonlinear eigenvalue problem on a finite interval, which exhibits a bifurcation from infinity and a non-smooth flat solution at a critical value, guiding the construction of subsolutions and the existence of a partially flat supersolution. The work proves existence of a global weak solution, nondegeneracy, and, via a parabolic comparison framework, uniqueness in the nondegenerate class, thereby rigorously justifying the formation of current wings near cathode edges and extending the one-dimensional Child-Langmuir theory to 2D with edge effects. These results provide a rigorous foundation for edge phenomena in space-charge-limited devices and offer a framework for further analysis of 2D electrostatic-flow problems with singular absorptions.

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 , separated to a finite distance When \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 ) must be found in order to get simultaneously Dirichlet and Neumann homogeneous boundary conditions in one of the extremes. If then the problem becomes much more difficult since the \textquotedblleft edge effects\textquotedblright\ arise in the plane and the electric current (now due to the presence of a very large perpendicular magnetic field) must be determined in order to get solutions of a singular semilinear equation which are partially flat ( 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 near the cathode edge to get such partially flat solutions.

Paper Structure

This paper contains 12 sections, 19 theorems, 243 equations, 11 figures.

Table of Contents

  1. Introduction
  2. On the modelling of the 2-d space-charge-limited flow
  3. The one-dimensional Child-Langmuir law revisited
  4. The autonomous case
  5. The distributed one-dimensional current density
  6. The 2-d parallel-plate geometry
  7. On the existence of solutions of $P_{a,b,j}$ via super and subsolutions method
  8. On the construction of some strict positive subsolutions with zero flux on $(-a,0)\times \{0\}$
  9. Bifurcation curve and flat solution for an auxiliary related nonlinear eigenvalue ordinary differential problem
  10. On the construction of the subsolution over $\Omega _{-}$: continuation
  11. Construction of a partially flat supersolution
  12. Proof of the uniqueness of nondegenerate solutions

Key Result

Theorem 1

There exists $A_{0},b_{0}>0$ and $\beta _{0}\in (0,\frac{1}{2}),$ such that, if $b\geq b_{0}>0,$and we assume with Then there exists a weak solution$u\in L^{2}(\Omega ;\delta )$ with$\delta =d((x,y),\partial \Omega ),$of problem $P_{a,b,j}$ and the additional conditions $AC_{a,b}$. Moreover and$Cy^{\alpha }\leq u(0,y),$$y\in (0,1),$ with $\alpha =\frac{2}{3}(2-\beta )$, $1<\overline{\alpha }\leq \

Figures (11)

  • Figure 1: Wings of $j(x)$ in some numerical experiments for different values of the cathode width $a$: adapted from Umstattd.
  • Figure 2: Bifurcation from infinity and critical values of $\lambda$ for a flat solution.
  • Figure 3: Progressive changes of profiles $u(\cdot,y).$
  • Figure 4: Boundary layer
  • Figure 5: Functions $f$ and $F.$
  • ...and 6 more figures

Theorems & Definitions (39)

  • Theorem 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Remark 8
  • Definition 9
  • Theorem 10
  • ...and 29 more