Bourgain–Brézis–Mironescu formula for magnetic operators

Abstract

Abstract. We prove a Bourgain-Brezis-Mironescu type formula for a class of nonlocal magnetic spaces, which builds a bridge between a fractional magnetic operator recently introduced and the classical theory.

Asymptotic Behaviours in Fractional Orlicz–Sobolev Spaces on Carnot Groups

“…Roughly speaking, these spaces are the natural functional setting in electromagnetism when dealing with particles interacting with a magnetic field. We refer to [44] for the analogous of (1) when p = 2, and to [42] for the general case and for magnetic BV functions. We finally refer to [37,38] for similar results for more general nonlocal functionals akin to those considered in [33].…”

Fractional magnetic Schrödinger‐Kirchhoff problems with convolution and critical nonlinearities

“…This paper was motivated by some works concerning the magnetic Schrödinger equation which have appeared in recent years (see other works) and have extensively studied , when the above magnetic As stated in Squassina and Volzone, up to correcting the operator by the factor (1− s ), it follows that converges to −(∇ u − iA ) 2 u as s →1.…”

“…Thus, up to normalization, the nonlocal case can be seen as an approximation of the local one. The motivation for its introduction was described in the literature and relies essentially on the Lévy‐Khintchine formula for the generator of a general Lévy process. If the magnetic field A ≡0, the operator can be reduced to the fractional Laplacian operator (−Δ) s , which may be viewed as the infinitesimal generator of a L vy stable diffusion processes .…”

Fractional NLS equations with magnetic field, critical frequency and critical growth

“…(iii) In [32] it was proved that, in the singular limit for α ր 1, the operator (1 − α)ε 2α (−∆) α Aε converges, in a suitable sense, to the classical local magnetic operator (1.4). Whence, up to multiplication by 1 − α the nonlocal theory is somehow consistent with the classical one.…”

“…The motivations for its introduction are described in [13,32] in more detail and rely essentially on the Lévy-Khintchine formula for the generator of a general Lévy process. If the magnetic field A ≡ 0, it seems that the first work which considered the existence of solutions for problem (1.1) in the subcritical case with ε = 1, formally α = 1 and K = 0 was [16].…”