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 nonlinear 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 regularization property of the operator (-Δ)-12. This also allows us to prove smoothness of the solutions. Our proof follows by direct arguments.

Borrelli, W., Multiple solutions for a self-consistent Dirac equation in two dimensions, <<JOURNAL OF MATHEMATICAL PHYSICS>>, 2018; 59 (4): N/A-N/A. [doi:10.1063/1.5005998] [http://hdl.handle.net/10807/171321]

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

Borrelli, W.
2018

Abstract

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 nonlinear 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 regularization property of the operator (-Δ)-12. This also allows us to prove smoothness of the solutions. Our proof follows by direct arguments.
Inglese
Borrelli, W., Multiple solutions for a self-consistent Dirac equation in two dimensions, <<JOURNAL OF MATHEMATICAL PHYSICS>>, 2018; 59 (4): N/A-N/A. [doi:10.1063/1.5005998] [http://hdl.handle.net/10807/171321]
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10807/171321
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 6
  • ???jsp.display-item.citation.isi??? 6
social impact