Abstract:This paper concerns mathematical theory of Meissner states of a bulk superconductor of type II, which occupies a bounded domain Ω in R 3 and is subjected to an applied magnetic field below the critical field HS. A Meissner state is described by a solution (f, A) of a nonlinear partial differential system called Meissner system, where f is a positive function on Ω which is equal to the modulus of the order parameter, and A is the magnetic potential defined on the entire space such that the inner trace of the no… Show more
“…The restriction for the domain to be simply-connected and without holes is removed in [35], hence the results in [10] remain true for a general bounded domain. More precise results of the location of concentration points is proved in [70], and the full model of Meissner states are studies in [44,50]. These results for (130) give the corresponding results for (129).…”
mentioning
confidence: 75%
“…We show that ζ is equal to a constant number. From (50) and 55, for any w ∈ W 1,p t0 (Ω, div 0) we have…”
Section: T0mentioning
confidence: 99%
“…In the same manner we define T φ,c for φ ∈ L p (Ω) and c ∈ R (see (50)). If furthermore φ has zero trace on ∂Ω then we may write T φ as a "gradient", T φ = ∇φ, which is understood as a functional on W 1,p t0 (Ω, R 3 ) .…”
Section: Setmentioning
confidence: 99%
“…For the Maxwell type equations, Schauder regularity of weak solutions has been established for many quasilinear models, see [10,46,47,35,50] for the model of Meissner states of superconductivity, [48,49] for the nonlinear Maxwell equations.…”
In this paper we revisit the nonlinear Maxwell system and Maxwell-Stokes system. One of the main feature of these systems is that existence of solutions depends not only on the natural of nonlinearity of the equations, but also on the type of the boundary conditions and the topology of the domain. We review and improve our recent results on existence of solutions by using the variational methods together with modified De Rham lemmas, and the operator methods. Regularity results by the reduction method are also discussed and improved.
“…The restriction for the domain to be simply-connected and without holes is removed in [35], hence the results in [10] remain true for a general bounded domain. More precise results of the location of concentration points is proved in [70], and the full model of Meissner states are studies in [44,50]. These results for (130) give the corresponding results for (129).…”
mentioning
confidence: 75%
“…We show that ζ is equal to a constant number. From (50) and 55, for any w ∈ W 1,p t0 (Ω, div 0) we have…”
Section: T0mentioning
confidence: 99%
“…In the same manner we define T φ,c for φ ∈ L p (Ω) and c ∈ R (see (50)). If furthermore φ has zero trace on ∂Ω then we may write T φ as a "gradient", T φ = ∇φ, which is understood as a functional on W 1,p t0 (Ω, R 3 ) .…”
Section: Setmentioning
confidence: 99%
“…For the Maxwell type equations, Schauder regularity of weak solutions has been established for many quasilinear models, see [10,46,47,35,50] for the model of Meissner states of superconductivity, [48,49] for the nonlinear Maxwell equations.…”
In this paper we revisit the nonlinear Maxwell system and Maxwell-Stokes system. One of the main feature of these systems is that existence of solutions depends not only on the natural of nonlinearity of the equations, but also on the type of the boundary conditions and the topology of the domain. We review and improve our recent results on existence of solutions by using the variational methods together with modified De Rham lemmas, and the operator methods. Regularity results by the reduction method are also discussed and improved.
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