On a class of semilinear evolution equations for vector potentials associated with Maxwell’s equations in Carnot groups

Franchi, Bruno; Obrecht, Enrico; Vecchi, Eugenio

doi:10.1016/j.na.2013.05.019

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Horizontal Magnetic Fields On the Heisenberg Group1

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2023

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Cited by 5 publications

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“…Then, formally as above, the magnetic (sub-)Laplacian in H 1 is given by (2.1) with the only difference that ∇u := (Xu)X + (Y u)Y is the now the horizontal gradient. Let us remark that the Rumin complex has already been applied to derive Maxwell's Equations in the more general setting of Carnot groups in [26,25].…”

Section: Horizontal Magnetic Fields On the Heisenberg Groupmentioning

confidence: 99%

Horizontal magnetic fields and improved Hardy inequalities in the Heisenberg group

Cassano¹,

Franceschi²,

Krejčiřı́k³

et al. 2021

Preprint

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In this paper we introduce a notion of magnetic field in the Heisenberg group and we study its influence on spectral properties of the corresponding magnetic (sub-elliptic) Laplacian. We show that uniform magnetic fields uplift the bottom of the spectrum. For magnetic fields vanishing at infinity, including Aharonov-Bohm potentials, we derive magnetic improvements to a variety of Hardy-type inequalities for the Heisenberg sub-Laplacian. In particular, we establish a sub-Riemannian analogue of Laptev and Weidl sub-criticality result for magnetic Laplacians in the plane. Instrumental for our argument is the validity of a Hardy-type inequality for the Folland-Stein operator, that we prove in this paper and has an interest on its own.

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Section: Horizontal Magnetic Fields On the Heisenberg Groupmentioning

confidence: 99%