Abstract In this paper we study a model which describes the relation of the matter and the electromagnetic field from a unitarian standpoint in the spirit of the ideas of Born and Infeld. This model, introduced in [1], is based on a semilinear perturbation of the Maxwell equation (SME). The particles are described by the finite energy solitary waves of SME whose existence is due to the presence of the nonlinearity. In the magnetostatic case (i.e. when the electric field E = 0 and the magnetic field H does not depend on time) the semilinear Maxwell equations reduce to the following semilinear equation ∇×(∇×A) = f (A) (1) where “∇×” is the curl operator, f is the gradient of a smooth function f :R3→R and A : R3 → R3 is the gauge potential related to the magnetic field H (H = ∇×A). The presence of the curl operator causes (1) to be a strongly degenerateelliptic equation. The existence of a nontrivial finite energy solution of (1) having a kind of cylindrical symmetry is proved. The proof is carried out by using a variational approach based on two main ingredients: the Principle of symmetric criticality of Palais, which allows to avoid the difficulties due to the curl operator, and the concentration-compactness argument combined with a suitable minimization argument.

Existence of static solutions of the semilinear Maxwell equations

FORTUNATO, Donato
2006

Abstract

Abstract In this paper we study a model which describes the relation of the matter and the electromagnetic field from a unitarian standpoint in the spirit of the ideas of Born and Infeld. This model, introduced in [1], is based on a semilinear perturbation of the Maxwell equation (SME). The particles are described by the finite energy solitary waves of SME whose existence is due to the presence of the nonlinearity. In the magnetostatic case (i.e. when the electric field E = 0 and the magnetic field H does not depend on time) the semilinear Maxwell equations reduce to the following semilinear equation ∇×(∇×A) = f (A) (1) where “∇×” is the curl operator, f is the gradient of a smooth function f :R3→R and A : R3 → R3 is the gauge potential related to the magnetic field H (H = ∇×A). The presence of the curl operator causes (1) to be a strongly degenerateelliptic equation. The existence of a nontrivial finite energy solution of (1) having a kind of cylindrical symmetry is proved. The proof is carried out by using a variational approach based on two main ingredients: the Principle of symmetric criticality of Palais, which allows to avoid the difficulties due to the curl operator, and the concentration-compactness argument combined with a suitable minimization argument.
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: http://hdl.handle.net/11586/19520
 Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus ND
  • ???jsp.display-item.citation.isi??? ND
social impact