2025/11/07 by Díaz, Jesús Ildefonso
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics
paper · doi:10.48550/arxiv.2511.05677
openalex publication_date 2025/11/07 · openalex created_date 2025/11/12 · openalex updated_date 2026/07/28
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≪ √(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>√(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=(∂ u)/(∂ 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.