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

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> , separated by a finite distance <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>D</mml:mi> </mml:math> . When <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>≪</mml:mo> <mml:msqrt> <mml:mi>A</mml:mi> </mml:msqrt> </mml:mrow> </mml:math> , “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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>j</mml:mi> </mml:math> ) must be found in order to get simultaneously Dirichlet and Neumann homogeneous boundary conditions in one of the extremes. If <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>&gt;</mml:mo> <mml:msqrt> <mml:mi>A</mml:mi> </mml:msqrt> </mml:mrow> </mml:math> , then the problem becomes much more difficult since the “edge effects” arise in the plane <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and the electric current (now <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>j</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> due to the presence of a very large perpendicular magnetic field) must be determined in order to get solutions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> of a singular semilinear equation which are partially flat ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mi>u</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mi>n</mml:mi> </mml:mrow> </mml:mfrac> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>j</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> near the cathode edge to get such partially flat solutions.