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Some characterizations of magnetic Sobolev spaces. (English) Zbl 1454.46034

Authors’ abstract: The aim of this note is to survey recent results contained in [H.-M. Nguyen and M. Squassina, Commun. Contemp. Math. 21, No. 1, Article ID 1850017, 13 p. (2019; Zbl 1418.46017); Adv. Nonlinear Anal. 7, No. 2, 227–245 (2018; Zbl 1429.49007); A. Pinamonti et al., Adv. Calc. Var. 12, No. 3, 225–252 (2019; Zbl 1433.46026); J. Math. Anal. Appl. 449, No. 2, 1152–1159 (2017; Zbl 1369.46031); M. Squassina and B. Volzone, C. R., Math., Acad. Sci. Paris 354, No. 8, 825–831 (2016; Zbl 1358.46033)], where the authors extended to the magnetic setting several characterizations of Sobolev and BV functions.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26A33 Fractional derivatives and integrals
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
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References:

[1] 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. · Zbl 1103.46310
[2] Davila, J., On an open question about functions of bounded variation, Calc Var Partial Differ Equ, 15, 519-527 (2002) · Zbl 1047.46025 · doi:10.1007/s005260100135
[3] Brézis, H., New approximations of the total variation and filters in imaging, Rend Accad Lincei, 26, 223-240 (2015) · Zbl 1325.26036
[4] Brézis, H.; Nguyen, H-M., The BBM formula revisited, Atti Accad Naz Lincei Rend Lincei Mat Appl, 27, 515-533 (2016) · Zbl 1362.46036 · doi:10.4171/RLM/746
[5] Brézis, H.; Nguyen, H-M., Two subtle convex nonlocal approximations of the BV-norm, Nonlinear Anal, 137, 222-245 (2016) · Zbl 1344.49017 · doi:10.1016/j.na.2016.02.005
[6] Maz’ya, V.; Shaposhnikova, T., On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J Funct Anal, 195, 230-238 (2002) · Zbl 1028.46050 · doi:10.1006/jfan.2002.3955
[7] Bourgain, J.; Nguyen, H-M., A new characterization of Sobolev spaces, C R Acad Sci Paris, 343, 75-80 (2006) · Zbl 1109.46034 · doi:10.1016/j.crma.2006.05.021
[8] Nguyen, H-M., Some new characterizations of Sobolev spaces, J Funct Anal, 237, 689-720 (2006) · Zbl 1109.46040 · doi:10.1016/j.jfa.2006.04.001
[9] Brézis, H.; Nguyen, H-M., Non-local functionals related to the total variation and connections with image processing, Ann PDE, 4, 9 (2018) · Zbl 1398.49042
[10] Nguyen, H-M., Γ-convergence, Sobolev norms, and BV functions, Duke Math J, 157, 495-533 (2011) · Zbl 1221.28011 · doi:10.1215/00127094-1272921
[11] Nguyen, H-M., Some inequalities related to Sobolev norms, Calc Var Partial Differ Equ, 41, 483-509 (2011) · Zbl 1226.46030 · doi:10.1007/s00526-010-0373-8
[12] Nguyen, H-M., Estimates for the topological degree and related topics, J Fixed Point Theory, 15, 185-215 (2014) · Zbl 1321.46037 · doi:10.1007/s11784-014-0182-3
[13] Nguyen, H-M; Squassina, M., Some remarks on rearrangement for nonlocal functionals, Nonlinear Anal, 162, 1-12 (2017) · Zbl 1384.46027 · doi:10.1016/j.na.2017.06.007
[14] Nguyen, H-M, Squassina, M.On anisotropic Sobolev spaces. Commun Contemp Math, to appear. DOI:10.1142/S0219199718500177 · Zbl 1418.46017
[15] Nguyen, H-M; Pinamonti, A.; Squassina, M., New characterizations of magnetic Sobolev spaces, Adv Nonlinear Anal, 7, 2, 227-245 (2018) · Zbl 1429.49007 · doi:10.1515/anona-2017-0239
[16] Pinamonti, A, Squassina, M, Vecchi, E.Magnetic BV functions and the Bourgain-Brezis-Mironescu formula. Adv Calc Var, to appear. DOI: · Zbl 1433.46026
[17] Pinamonti, A.; Squassina, M.; Vecchi, E., The Maz’ya-Shaposhnikova limit in the magnetic setting, J Math Anal Appl, 449, 1152-1159 (2017) · Zbl 1369.46031 · doi:10.1016/j.jmaa.2016.12.065
[18] Squassina, M.; Volzone, B., Bourgain-Brezis-Mironescu formula for magnetic operators, C R Math Acad Sci Paris, 354, 825-831 (2016) · Zbl 1358.46033 · doi:10.1016/j.crma.2016.04.013
[19] Avron, J.; Herbst, I.; Simon, B., Schrödinger operators with magnetic fields. I. General interactions, Duke Math J, 45, 847-883 (1978) · Zbl 0399.35029 · doi:10.1215/S0012-7094-78-04540-4
[20] Reed, M.; Simon, B., Methods of modern mathematical physics, I, functional analysis (1980), New York: Academic Press, Inc., New York · Zbl 0459.46001
[21] Arioli, G.; Szulkin, A., A semilinear Schrödinger equation in the presence of a magnetic field, Arch Ration Mech Anal, 170, 277-295 (2003) · Zbl 1051.35082 · doi:10.1007/s00205-003-0274-5
[22] 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). · Zbl 0702.35067
[23] Squassina, M., Soliton dynamics for the nonlinear Schrödinger equation with magnetic field, Manuscripta Math, 130, 461-494 (2009) · Zbl 1179.81066 · doi:10.1007/s00229-009-0307-y
[24] Lieb, EH, Loss, M.Analysis. Providence (RI): American Mathematical Society; 2001. (Graduate studies in mathematics; 14). · Zbl 0966.26002
[25] Di Nezza, E.; Palatucci, G.; Valdinoci, E., Hitchhiker’s guide to the fractional Sobolev spaces, Bull Sci Math, 136, 521-573 (2012) · Zbl 1252.46023 · doi:10.1016/j.bulsci.2011.12.004
[26] 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). · Zbl 1266.81079
[27] Binlin, Z.; Squassina, M.; Xia, Z., Fractional NLS equations with magnetic field, critical frequency and critical growth, Manuscripta Math, 155, 1-2, 115-140 (2018) · Zbl 1401.35311 · doi:10.1007/s00229-017-0937-4
[28] d’Avenia, P.; Squassina, M., Ground states for fractional magnetic operators, ESAIM COCV, 24, 1, 1-24 (2018) · Zbl 1400.49059 · doi:10.1051/cocv/2016071
[29] Fiscella, A.; Pinamonti, A.; Vecchi, E., Multiplicity results for magnetic fractional problems, J Differ Equ, 263, 8, 4617-4633 (2017) · Zbl 1432.35220 · doi:10.1016/j.jde.2017.05.028
[30] Liang, S.; Repovs, D.; Zhang, B., On the fractional Schrödinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity, Comput Math Appl, 75, 1778-1794 (2018) · Zbl 1409.78002 · doi:10.1016/j.camwa.2017.11.033
[31] Mingqi, X.; Pucci, P.; Squassina, M., Nonlocal Schrödinger-Kirchhoff equations with external magnetic field, Discrete Contin Dyn Syst A, 37, 503-521 (2017)
[32] Mingqi, X, Radulescu, V, B. Zhang, V.A critical fractional Choquard-Kirchhoff problem with magnetic field. Commun Contemp Math. 2018, to appear. DOI: · Zbl 1416.49012
[33] Wang, F.; Xiang, M., Multiplicity of solutions to a nonlocal Choquard equation involving fractional magnetic operators and critical exponent, Electron J Differ Equ, 2016, 1-11 (2016) · Zbl 1418.92147 · doi:10.1186/s13662-015-0739-5
[34] Ambrosio, L.; Fusco, N.; Pallara, D., Functions of bounded variation and free discontinuity problems (2000), New York: The Clarendon Press, Oxford University Press, New York · Zbl 0957.49001
[35] Brézis, H., How to recognize constant functions. Connections with Sobolev spaces, Russian Math Surveys, 57, 693-708 (2002) · Zbl 1072.46020 · doi:10.1070/RM2002v057n04ABEH000533
[36] Bourgain, J.; Brézis, H.; Mironescu, P., Limiting embedding theorems for \(####\) when \(####\) and applications, J Anal Math, 87, 77-101 (2002) · Zbl 1029.46030 · doi:10.1007/BF02868470
[37] Ponce, A., A new approach to Sobolev spaces and connections to Γ-convergence, Calc Var Partial Differ Equ, 19, 229-255 (2004) · Zbl 1352.46037 · doi:10.1007/s00526-003-0195-z
[38] Ponce, A.; Spector, D., On formulae decoupling the total variation of BV functions, Nonlinear Anal, 154, 241-257 (2017) · Zbl 1359.26011 · doi:10.1016/j.na.2016.08.028
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