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Multiple solutions for a self-consistent Dirac equation in two dimensions

Borrelli, William (2018), Multiple solutions for a self-consistent Dirac equation in two dimensions, Journal of Mathematical Physics, 59, 4. 10.1063/1.5005998

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Type
Article accepté pour publication ou publié
Date
2018
Journal name
Journal of Mathematical Physics
Volume
59
Number
4
Publisher
American Institute of Physics
Publication identifier
10.1063/1.5005998
Metadata
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Author(s)
Borrelli, William
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)
This paper is devoted to the variational study of an effective model for the electron transport in a graphene sample. We prove the existence of infinitely many stationary solutions for a nonlin-ear Dirac equation which appears in the WKB limit for the Schrödinger equation describing the semi-classical electron dynamics. The interaction term is given by a mean field, self-consistent potential which is the trace of the 3D Coulomb potential. Despite the nonlinearity being 4-homogeneous, compactness issues related to the limiting Sobolev embedding H12(Ω,C)→L4(Ω,C) are avoided thanks to the regular-ization property of the operator $(-\Delta)^{-\frac{1}{2}$. This also allows us to prove smoothness of the solutions. Our proof follows by direct arguments.
Subjects / Keywords
electron transport; Graphene; Nonlinear Dirac equation; Self-consistent model

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