We study a class of minimization problems for a nonlocal operator involving an external magnetic potential. The notions are physically justified and consistent with the case of absence of magnetic fields. Existence of solutions is obtained via concentration compactness.
Accepté le :
DOI : 10.1051/cocv/2016071
Mots-clés : Fractional magnetic operators, minimization problems, concentration compactness
@article{COCV_2018__24_1_1_0,
author = {d{\textquoteright}Avenia, Pietro and Squassina, Marco},
title = {Ground states for fractional magnetic operators},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1--24},
publisher = {EDP-Sciences},
volume = {24},
number = {1},
year = {2018},
doi = {10.1051/cocv/2016071},
mrnumber = {3764131},
zbl = {1400.49059},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv/2016071/}
}
TY - JOUR AU - d’Avenia, Pietro AU - Squassina, Marco TI - Ground states for fractional magnetic operators JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 1 EP - 24 VL - 24 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2016071/ DO - 10.1051/cocv/2016071 LA - en ID - COCV_2018__24_1_1_0 ER -
%0 Journal Article %A d’Avenia, Pietro %A Squassina, Marco %T Ground states for fractional magnetic operators %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 1-24 %V 24 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2016071/ %R 10.1051/cocv/2016071 %G en %F COCV_2018__24_1_1_0
d’Avenia, Pietro; Squassina, Marco. Ground states for fractional magnetic operators. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 1, pp. 1-24. doi : 10.1051/cocv/2016071. http://archive.numdam.org/articles/10.1051/cocv/2016071/
A. Applebaum, Lévy processes and Stochastic Calculus. In vol. 116 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2009). | MR | Zbl
and , A semilinear Schrödinger equation in the presence of a magnetic field. Arch. Ration. Mech. Anal. 170 (2003) 277–295. | DOI | MR | Zbl
, and , Schrödinger operators with magnetic fields. I. General interactions. Duke Math. J. 45 (1978) 847–883. | DOI | MR | Zbl
and , The second eigenvalue of the fractional -Laplacian. Adv. Calc. Var. 9 (2016) 323–355. | DOI | MR | Zbl
J. Bourgain, H. Brezis and P. Mironescu, Another look at Sobolev spaces, in Optimal Control and Partial Differential Equations. A Volume in Honor of Professor Alain Bensoussan’s 60th Birthday, edited by J.L. Menaldi, E. Rofman and A. Sulem. IOS Press, Amsterdam (2001) 439–455. | MR | Zbl
, and , Limiting embedding theorems for when and applications. J. Anal. Math. 87 (2002) 77–101. | DOI | MR | Zbl
, and , Classification of solutions for an integral equation. Commun. Pure Appl. Math. 59 (2006) 330–343. | DOI | MR | Zbl
and , Semiclassical limit for nonlinear Schrödinger equations with electromagnetic fields. J. Math. Anal. Appl. 275 (2002) 108–130. | DOI | MR | Zbl
R. Cont and P. Tankov, Financial modeling with jump processes. Financial Mathematics Series. Chapman & Hall/CRC, Boca Raton, FL (2004). | MR | Zbl
and , Best constants for Sobolev inequalities for higher order fractional derivatives. J. Math. Anal. Appl. 295 (2004) 225–236. | DOI | MR | Zbl
P. d’Avenia, and , On fractional Choquard equations. Math. Models Methods Appl. Sci. 25 (2015) 1447–1476. | DOI | MR | Zbl
and , Semiclassical stationary states for nonlinear Schrödinger equations under a strong external magnetic field. J. Differ. Equ. 259 (2015) 596–627. | DOI | MR | Zbl
, and , Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136 (2012) 521–573. | DOI | MR | Zbl
, and , Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian. Matematiche 68 (2013) 201–216. | MR | Zbl
M. Esteban and P.L. Lions, Stationary solutions of nonlinear Schrödinger equations with an external magnetic field, Partial differential equations and the calculus of variations. Vol. I of Progr. Nonlinear Differential Equations Appl. Birkhäuser Boston, Boston, MA (1989) 401–449. | MR | Zbl
, and , Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators. J. Amer. Math. Soc. 21 (2008) 925–950. | DOI | MR | Zbl
, Essential selfadjointness of the Weyl quantized relativistic Hamiltonian. Ann. Inst. Henri Poincaré, Phys. Théor. 51 (1989) 265–297. | Numdam | MR | Zbl
T. Ichinose, Magnetic relativistic Schrödinger operators and imaginary-time path integrals, Mathematical physics, spectral theory and stochastic analysis. Oper. Theory Adv. Appl. In vol. 232 of Birkhäuser/Springer Basel AG, Basel (2013) 247–297. | MR | Zbl
and , Imaginary-time path integral for a relativistic spinless particle in an electromagnetic field. Commut. Math. Phys. 105 (1986) 239–257. | DOI | MR | Zbl
, and , Magnetic pseudodifferential operators. Publ. Res. Inst. Math. Sci. 43 (2007) 585–623. | DOI | MR | Zbl
, Existence and semi-classical limit of the least energy solution to a nonlinear Schrödinger equation with electromagnetic fields. Nonlin. Anal. 41 (2000) 763–778. | DOI | MR | Zbl
, Fractional Schrödinger equation. Phys. Rev. E 66 (2002) 056108. | DOI | MR
E.H. Lieb and R. Seiringer, The stability of matter in quantum mechanics. Cambridge University Press, Cambridge (2010). | MR | Zbl
, Symétrie and compacité dans le espaces de Sobolev. J. Funct. Anal. 49 (1982) 315–334. | DOI | MR | Zbl
, The concentration-compactness principle in the calculus of variation. The locally compact case I. Ann. Inst. Henri Poincaré Anal. Non Linéaire 1 (1984) 109–145. | DOI | Numdam | MR | Zbl
and , The restaurant at the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A 37 (2004) R161–R208. | DOI | MR | Zbl
M. Reed, B. Simon, Methods of modern mathematical physics, I, Functional analysis. Academic Press, Inc., New York (1980). | MR | Zbl
, Soliton dynamics for the nonlinear Schrödinger equation with magnetic field. Manuscr. Math. 130 (2009) 461–494. | DOI | MR | Zbl
Cité par Sources :
