[Faisceaux d’électrons : solutions partiellement plates d’une équation elliptique non linéaire avec un terme d’absorption singulier]
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 by a finite distance $D$. When $D\ll \sqrt{A}$, “edge effects” 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 “edge effects” 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 Rokhlenko (2003) and Alexander Rokhlenko (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.
Dans la loi dite de Child–Langmuir, établie depuis 1911, un faisceau d’électrons est formé reliant deux électrodes, supposées être deux plaques parallèles d’aire $A$, séparées d’une distance finie $D$. Lorsque $D\ll \sqrt{A}$ les « effets de bord » sont négligeables et la modélisation se réduit à un problème aux limites non linéaire pour une équation différentielle ordinaire singulière dans laquelle un coefficient constant (le courant électrique généré $j$) doit être trouvé afin d’obtenir simultanément des conditions aux limites homogènes de Dirichlet et de Neumann dans l’un des extrêmes. Si $D>\sqrt{A}$, alors le problème devient beaucoup plus difficile car les « effets de bord » apparaissent dans le plan $(x,y)$ et le courant électrique (maintenant $j(x)$ en raison de la présence d’un champ magnétique perpendiculaire très important) doit être déterminé afin d’obtenir des solutions $u(x,y)$ d’une équation semi-linéaire singulière qui soient partiellement plates ($u = \frac{\partial u}{\partial n} = 0$ sur une partie du bord). Dans cet article, nous proposons un traitement mathématique rigoureux de certaines études antérieures (Joel Lebowitz et Alexander Rokhlenko (2003) et Alexander Rokhlenko (2006)), où plusieurs questions ouvertes ont été laissées ouvertes : par exemple, la nécessité d’une singularité de $j(x)$ près du bord de la cathode pour obtenir de telles solutions partiellement plates.
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Keywords: Electron beams, space charge, Child–Langmuir law, singular semilinear equation, partially flat solution, super and subsolution method, $H^1$-matching and its generalizations
Mots-clés : Faisceaux d’électrons, charge de l’espace, loi de Child–Langmuir, équation semi-linéaire singulière, solution partiellement plate, méthode de super et sous-solution, couplage $H^1$ et ses généralisations
Jesús Ildefonso Díaz 1
CC-BY 4.0
@article{CRMATH_2025__363_G12_1301_0,
author = {Jes\'us Ildefonso D{\'\i}az},
title = {Electron beams: partially flat solutions of a nonlinear elliptic equation with a singular absorption term},
journal = {Comptes Rendus. Math\'ematique},
pages = {1301--1338},
year = {2025},
publisher = {Acad\'emie des sciences, Paris},
volume = {363},
doi = {10.5802/crmath.792},
language = {en},
}
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%0 Journal Article %A Jesús Ildefonso Díaz %T Electron beams: partially flat solutions of a nonlinear elliptic equation with a singular absorption term %J Comptes Rendus. Mathématique %D 2025 %P 1301-1338 %V 363 %I Académie des sciences, Paris %R 10.5802/crmath.792 %G en %F CRMATH_2025__363_G12_1301_0
Jesús Ildefonso Díaz. Electron beams: partially flat solutions of a nonlinear elliptic equation with a singular absorption term. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1301-1338. doi: 10.5802/crmath.792
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