Some characterizations of magnetic Sobolev spaces

References

• Bourgain J, Brézis H, Mironescu P. Another look at Sobolev spaces. In Menaldi JL, Rofman E, and Sulem A, editors. Optimal control and partial differential equations. A volume in honor of Professor Alain Bensoussan's 60th birthday. Amsterdam: IOS Press; 2001. p. 439–455.
   Google Scholar
• Davila J. On an open question about functions of bounded variation. Calc Var Partial Differ Equ. 2002;15:519–527. doi: 10.1007/s005260100135
   Web of Science ®Google Scholar
• Brézis H. New approximations of the total variation and filters in imaging. Rend Accad Lincei. 2015;26:223–240.
   Google Scholar
• Brézis H, Nguyen H-M. The BBM formula revisited. Atti Accad Naz Lincei Rend Lincei Mat Appl. 2016;27:515–533. doi: 10.4171/RLM/746
   Google Scholar
• Brézis H, Nguyen H-M. Two subtle convex nonlocal approximations of the BV-norm. Nonlinear Anal. 2016;137:222–245. doi: 10.1016/j.na.2016.02.005
   Web of Science ®Google Scholar
• Maz'ya V, Shaposhnikova T. On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces. J Funct Anal. 2002;195:230–238. Erratum to: On the Bourgain, Brezis and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces. J Funct Anal. 2003;201:298–300. doi: 10.1006/jfan.2002.3955
   Web of Science ®Google Scholar
• Bourgain J, Nguyen H-M. A new characterization of Sobolev spaces. C R Acad Sci Paris. 2006;343:75–80. doi: 10.1016/j.crma.2006.05.021
   Google Scholar
• Nguyen H-M. Some new characterizations of Sobolev spaces. J Funct Anal. 2006;237:689–720. doi: 10.1016/j.jfa.2006.04.001
   Web of Science ®Google Scholar
• Brézis H, Nguyen H-M. Non-local functionals related to the total variation and connections with image processing. Ann PDE. 2018;4:9. Available from: https://doi.org/10.1007/s40818-018-0044-1
   Google Scholar
• Nguyen H-M. Γ-convergence, Sobolev norms, and BV functions. Duke Math J. 2011;157:495–533. doi: 10.1215/00127094-1272921
   Web of Science ®Google Scholar
• Nguyen H-M. Some inequalities related to Sobolev norms. Calc Var Partial Differ Equ. 2011;41:483–509. doi: 10.1007/s00526-010-0373-8
   Web of Science ®Google Scholar
• Nguyen H-M. Estimates for the topological degree and related topics. J Fixed Point Theory. 2014;15:185–215. doi: 10.1007/s11784-014-0182-3
   Web of Science ®Google Scholar
• Nguyen H-M, Squassina M. Some remarks on rearrangement for nonlocal functionals. Nonlinear Anal. 2017;162:1–12. doi: 10.1016/j.na.2017.06.007
   Web of Science ®Google Scholar
• Nguyen H-M, Squassina M. On anisotropic Sobolev spaces. Commun Contemp Math, to appear. DOI:10.1142/S0219199718500177
   Google Scholar
• Nguyen H-M, Pinamonti A, Squassina M, et al. New characterizations of magnetic Sobolev spaces. Adv Nonlinear Anal. 2018;7(2):227–245. doi: 10.1515/anona-2017-0239
   Web of Science ®Google Scholar
• Pinamonti A, Squassina M, Vecchi E. Magnetic BV functions and the Bourgain-Brezis-Mironescu formula. Adv Calc Var, to appear. DOI:10.1515/acv-2017-0019
   Google Scholar
• Pinamonti A, Squassina M, Vecchi E. The Maz'ya-Shaposhnikova limit in the magnetic setting. J Math Anal Appl. 2017;449:1152–1159. doi: 10.1016/j.jmaa.2016.12.065
   Web of Science ®Google Scholar
• Squassina M, Volzone B. Bourgain-Brezis-Mironescu formula for magnetic operators. C R Math Acad Sci Paris. 2016;354:825–831. doi: 10.1016/j.crma.2016.04.013
   Google Scholar
• Avron J, Herbst I, Simon B. Schrödinger operators with magnetic fields. I. General interactions. Duke Math J. 1978;45:847–883. doi: 10.1215/S0012-7094-78-04540-4
   Web of Science ®Google Scholar
• Reed M, Simon B. Methods of modern mathematical physics, I, functional analysis. New York: Academic Press, Inc.; 1980.
   Google Scholar
• Arioli G, Szulkin A. A semilinear Schrödinger equation in the presence of a magnetic field. Arch Ration Mech Anal. 2003;170:277–295. doi: 10.1007/s00205-003-0274-5
   Web of Science ®Google Scholar
• Esteban M, Lions P-L. Stationary solutions of nonlinear Schrödinger equations with an external magnetic field. In Partial differential equations and the calculus of variations. Vol. I. Boston (MA): Birkhäuser Boston; 1989. p. 401–449. (Progr. nonlinear differential equations appl.; 1).
   Google Scholar
• Squassina M. Soliton dynamics for the nonlinear Schrödinger equation with magnetic field. Manuscripta Math. 2009;130:461–494. doi: 10.1007/s00229-009-0307-y
   Web of Science ®Google Scholar
• Lieb EH, Loss M. Analysis. Providence (RI): American Mathematical Society; 2001. (Graduate studies in mathematics; 14).
   Google Scholar
• Di Nezza E, Palatucci G, Valdinoci E. Hitchhiker's guide to the fractional Sobolev spaces. Bull Sci Math. 2012;136:521–573. doi: 10.1016/j.bulsci.2011.12.004
   Web of Science ®Google Scholar
• Ichinose T. Magnetic relativistic Schrödinger operators and imaginary-time path integrals. In: Mathematical physics, spectral theory and stochastic analysis. Basel: Birkhäuser/Springer; 2013. p. 247–297. (Oper. theory adv. appl.; 232).
   Google Scholar
• Binlin Z, Squassina M, Xia Z. Fractional NLS equations with magnetic field, critical frequency and critical growth. Manuscripta Math. 2018;155(1–2):115–140. doi: 10.1007/s00229-017-0937-4
   Web of Science ®Google Scholar
• d'Avenia P, Squassina M. Ground states for fractional magnetic operators. ESAIM COCV. 2018;24(1):1–24. doi: 10.1051/cocv/2016071
   Web of Science ®Google Scholar
• Fiscella A, Pinamonti A, Vecchi E. Multiplicity results for magnetic fractional problems. J Differ Equ. 2017;263(8):4617–4633. doi: 10.1016/j.jde.2017.05.028
   Web of Science ®Google Scholar
• Liang S, Repovs D, Zhang B. On the fractional Schrödinger–Kirchhoff equations with electromagnetic fields and critical nonlinearity. Comput Math Appl. 2018;75:1778–1794. doi: 10.1016/j.camwa.2017.11.033
   Web of Science ®Google Scholar
• Mingqi X, Pucci P, Squassina M, et al. Nonlocal Schrödinger-Kirchhoff equations with external magnetic field. Discrete Contin Dyn Syst A. 2017;37:503–521.
   Web of Science ®Google Scholar
• Mingqi X, Radulescu V, B. Zhang V. A critical fractional Choquard–Kirchhoff problem with magnetic field. Commun Contemp Math. 2018, to appear. DOI:10.1142/S0219199718500049
   Google Scholar
• Wang F, Xiang M. Multiplicity of solutions to a nonlocal Choquard equation involving fractional magnetic operators and critical exponent. Electron J Differ Equ. 2016;2016:1–11. doi: 10.1186/s13662-015-0739-5
   Google Scholar
• Ambrosio L, Fusco N, Pallara D. Functions of bounded variation and free discontinuity problems. New York: The Clarendon Press, Oxford University Press; 2000.
   Google Scholar
• Brézis H. How to recognize constant functions. Connections with Sobolev spaces. Russian Math Surveys. 2002;57:693–708. doi: 10.1070/RM2002v057n04ABEH000533
   Web of Science ®Google Scholar
• Bourgain J, Brézis H, Mironescu P. Limiting embedding theorems for Ws,p when s↑1 and applications. J Anal Math. 2002;87:77–101. doi: 10.1007/BF02868470
   Web of Science ®Google Scholar
• Ponce A. A new approach to Sobolev spaces and connections to Γ-convergence. Calc Var Partial Differ Equ. 2004;19:229–255. doi: 10.1007/s00526-003-0195-z
   Web of Science ®Google Scholar
• Ponce A, Spector D. On formulae decoupling the total variation of BV functions. Nonlinear Anal. 2017;154:241–257. doi: 10.1016/j.na.2016.08.028
   Web of Science ®Google Scholar