Hitchhikerʼs guide to the fractional Sobolev spaces
Tài liệu tham khảo
Adams, 1975
Alberti, 1998, Phase transition with the line-tension effect, Arch. Ration. Mech. Anal., 144, 1, 10.1007/s002050050111
Aronszajn, 1955, Boundary values of functions with finite Dirichlet integral, Tech. Report of Univ. of Kansas, 14, 77
Bates, 2006, On some nonlocal evolution equations arising in materials science, vol. 48, 13
Beckner, 1975, Inequalities in Fourier analysis in Rn, Proc. Nat. Acad. Sci. USA, 72, 638, 10.1073/pnas.72.2.638
Besov, 1959, On a certain family of functional spaces. Imbedding and continuation theorems, Dokl. Akad. Nauk SSSR, 126, 1163
Besov, 1959, On some conditions of membership in Lp for derivatives of periodic functions, Naučn. Dokl. Vyss. Skoly. Fiz.-Mat. Nauki, 13
Biler, 2010, Nonlinear diffusion of dislocation density and self-similar solutions, Comm. Math. Phys., 294, 145, 10.1007/s00220-009-0855-8
Biler, 2001, Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18, 613, 10.1016/S0294-1449(01)00080-4
Brezis, 2002, How to recognize constant functions. Connections with Sobolev spaces, Uspekhi Mat. Nauk, 57, 59
Bourgain, 2001, Another look at Sobolev spaces, 439
Brezis, 2001, Gagliardo–Nirenberg, composition and products in fractional Sobolev spaces, J. Evol. Equ., 1, 387, 10.1007/PL00001378
Cabré, 2010, Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian, Discrete Contin. Dyn. Syst., 28, 1179, 10.3934/dcds.2010.28.1179
Cabré, 2005, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math., 58, 1678, 10.1002/cpa.20093
L. Caffarelli, A. Mellet, Y. Sire, Traveling waves for a boundary reaction–diffusion equation, preprint, available online at http://arxiv.org/abs/1101.4381v1.
Caffarelli, 2010, Non-local minimal surfaces, Comm. Pure Appl. Math., 63, 1111
Caffarelli, 2010, Variational problems with free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12, 1151, 10.4171/JEMS/226
Caffarelli, 2008, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171, 425, 10.1007/s00222-007-0086-6
Caffarelli, 2007, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32, 1245, 10.1080/03605300600987306
Caffarelli, 2011, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations, 41, 203, 10.1007/s00526-010-0359-6
Caffarelli, 2010, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math., 171, 1903, 10.4007/annals.2010.171.1903
Campanato, 1963, Proprietà di hölderianità di alcune classi di funzioni, Ann. Sc. Norm. Super. Pisa, 17, 175
Chermisi, 2009, Fibered nonlinearities for p(x)-Laplace equations, Adv. Calc. Var., 2, 185, 10.1515/ACV.2009.008
Chermisi, 2010, A symmetry result for a general class of divergence form PDEs in fibered media, Nonlinear Anal., 73, 695, 10.1016/j.na.2010.04.002
Colton, 1998, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93
Cont, 2004, Financial Modelling with Jump Processes
Cordoba, 1998, Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation, Ann. of Math. (2), 148, 1135, 10.2307/121037
Cotsiolis, 2004, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295, 225, 10.1016/j.jmaa.2004.03.034
Craig, 1994, Hamiltonian long-wave approximations to the water-wave problem, Wave Motion, 19, 367, 10.1016/0165-2125(94)90003-5
Craig, 2000, Travelling two and three dimensional capillary gravity water waves, SIAM J. Math. Anal., 32, 323, 10.1137/S0036141099354181
Craig, 1997, The modulational regime of three-dimensional water waves and the Davey–Stewartson system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14, 615, 10.1016/S0294-1449(97)80128-X
Craig, 1992, Nonlinear modulation of gravity waves: a rigorous approach, Nonlinearity, 5, 497, 10.1088/0951-7715/5/2/009
Craig, 1995, An integrable normal form for water waves in infinite depth, Phys. D, 84, 513, 10.1016/0167-2789(95)00067-E
de la Llave, 1986, Gravity waves on the surface of the sphere, J. Nonlinear Sci., 6, 147, 10.1007/BF02434052
de la Llave, 2009, Symmetry for a Dirichlet–Neumann problem arising in water waves, Math. Res. Lett., 16, 909, 10.4310/MRL.2009.v16.n5.a13
Duistermaat, 1975, The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math., 29, 39, 10.1007/BF01405172
Duvaut, 1976, Inequalities in Mechanics and Physics, vol. 219
Dyda, 2010, Embedding theorems for Lipschitz and Lorentz spaces on lower Ahlfors regular sets, Studia Math., 197, 247, 10.4064/sm197-3-3
Esposito, 2004, Sharp regularity for functionals with (p,q) growth, J. Differential Equations, 204, 5, 10.1016/j.jde.2003.11.007
A. Farina, E. Valdinoci, Rigidity results for elliptic PDEs with uniform limits: an abstract framework with applications, Indiana Univ. Math. J., in press, available online at http://www.iumj.indiana.edu/IUMJ/Preprints/4433.pdf.
Fefferman, 1986, Relativistic stability of matter. I, Rev. Mat. Iberoam., 2, 119, 10.4171/RMI/30
Gächter, 2003, Dirichlet-to-Neumann map for three-dimensional elastic waves, Wave Motion, 37, 293, 10.1016/S0165-2125(02)00091-4
Gagliardo, 1957, Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili, Rend. Sem. Mat. Univ. Padova, 27, 284
Gagliardo, 1958, Proprietà di alcune classi di funzioni in più variabili, Ric. Mat., 7, 102
Garroni, 2006, A singular perturbation result with a fractional norm, vol. 68, 111
Giusti, 1994
Gonzalez, 2012, Slow motion of particle systems as a limit of a reaction–diffusion equation with half-Laplacian in dimension one, Discrete Cont. Dyn. Syst. A, 32, 1255, 10.3934/dcds.2012.32.1255
Gilbarg, 2001
Grote, 2004, Dirichlet-to-Neumann boundary conditions for multiple scattering problems, J. Comput. Phys., 201, 630, 10.1016/j.jcp.2004.06.012
Hu, 2005, Analyticity of Dirichlet–Neumann operators on Hölder and Lipschitz domains, SIAM J. Math. Anal., 37, 302, 10.1137/S0036141004444810
Ishii, 2010, A class of integral equations and approximation of p-laplace equations, Calc. Var. Partial Differential Equations, 37, 485, 10.1007/s00526-009-0274-x
Johnson, 1985, Book review: Theory of function spaces, Bull. Amer. Math. Soc., 13, 76, 10.1090/S0273-0979-1985-15376-5
Karlsen, 2011, A duality approach to the fractional laplacian with measure data, Publ. Mat., 55, 151, 10.5565/PUBLMAT_55111_07
Kolmogorov, 1931, Über Kompaktheit der Funktionenmengen bei der Konvergenz im Mittel, Nachr. Ges. Wiss. Göttingen, 9, 60
Kristensen, 2006, The singular set of minima of integral functionals, Arch. Ration. Mech. Anal., 180, 331, 10.1007/s00205-005-0402-5
Kurzke, 2006, A nonlocal singular perturbation problem with periodic well potential, ESAIM Control Optim. Calc. Var., 12, 52, 10.1051/cocv:2005037
Jones, 1981, Quasiconformal mappings and extendability of functions in Sobolev spaces, Acta Math., 147, 71, 10.1007/BF02392869
Landkof, 1973, Foundations of Modern Potential Theory, vol. 180
Leoni, 2009, A First Course in Sobolev Spaces, vol. 105
Lions, 1963, Théorèmes de trace et dʼinterpolation (IV), Math. Ann., 151, 42, 10.1007/BF01343324
Lions, 1968, Problémes aux limites non homogénes et applications, vol. 1, vol. 17
Lizorkin, 1960, Boundary properties of functions from “weight” classes, Dokl. Akad. Nauk SSSR, 132, 514
Majda, 1996, A two-dimensional model for quasigeostrophic flow: comparison with the two-dimensional Euler flow, Phys. D, 98, 515, 10.1016/0167-2789(96)00114-5
Mellet, 2011, Fractional diffusion limit for collisional kinetic equations, Arch. Ration. Mech. Anal., 199, 493, 10.1007/s00205-010-0354-2
Mazʼya, 2002, On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., 195, 230, 10.1006/jfan.2002.3955
Mazʼya, 2009, Theory of Sobolev Multipliers. With Applications to Differential and Integral Operators, vol. 337
Metzler, 2000, The random walkʼs guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339, 1, 10.1016/S0370-1573(00)00070-3
Milakis, 2008, Regularity for the nonlinear Signorini problem, Adv. Math., 217, 1301, 10.1016/j.aim.2007.08.009
Mingione, 2003, The singular set of solutions to non-differentiable elliptic systems, Arch. Ration. Mech. Anal., 166, 287, 10.1007/s00205-002-0231-8
Mingione, 2007, The Calderón–Zygmund theory for elliptic problems with measure data, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 6, 195
Mingione, 2011, Gradient potential estimates, J. Eur. Math. Soc., 13, 459
Naumkin, 1994
Nicholls, 2008, Joint analyticity and analytic continuation of Dirichlet–Neumann operators on doubly perturbed domains, J. Math. Fluid Mech., 10, 238, 10.1007/s00021-006-0231-9
G. Palatucci, A. Pisante, Sobolev embeddings and concentration-compactness alternative for fractional Sobolev spaces, submitted for publication, available online at http://mipa.unimes.fr/preprints.html.
G. Palatucci, O. Savin, E. Valdinoci, Local and global minimizers for a variational energy involving a fractional norm, Ann. Mat. Pura Appl., in press, doi:10.1007/s10231-011-0243-9.
Palatucci, 2010, Γ-convergence of some super quadratic functionals with singular weights, Math. Z., 266, 533, 10.1007/s00209-009-0584-x
Riesz, 1933, Sur les ensembles compacts de fonctions sommables, Acta Szeged Sect. Math., 6, 136
Runst, 1996, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, vol. 3
Stoker, 1957, Water Waves: The Mathematical Theory with Applications, vol. IV
O. Savin, Personal communication, 2011.
Savin, 2009, Elliptic PDEs with fibered nonlinearities, J. Geom. Anal., 19, 420, 10.1007/s12220-008-9064-5
Savin, 2011, Density estimates for a nonlocal variational model via the Sobolev inequality, SIAM J. Math. Anal., 43, 2675, 10.1137/110831040
O. Savin, E. Valdinoci, Density estimates for a variational model driven by the Gagliardo norm, submitted for publication, available online at http://arxiv.org/abs/1007.2114v3.
L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, PhD thesis, Austin University, 2005, available online at http://www.math.uchicago.edu/~luis/preprints/luisdissreadable.pdf.
Silvestre, 2007, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60, 67, 10.1002/cpa.20153
Sire, 2009, Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, J. Funct. Anal., 256, 1842, 10.1016/j.jfa.2009.01.020
Slobodeckij, 1958, Generalized Sobolev spaces and their applications to boundary value problems of partial differential equations, Leningrad. Gos. Ped. Inst. Učep. Zap., 197, 54
Stein, 1970, Singular Integrals and Differentiability Properties of Functions, vol. 30
Tartar, 2007, An Introduction to Sobolev Spaces and Interpolation Spaces, vol. 3
Toland, 1997, The Peierls–Nabarro and Benjamin–Ono equations, J. Funct. Anal., 145, 136, 10.1006/jfan.1996.3016
Triebel, 1977, Interpolation Theory, Fourier Analysis and Function Spaces
Triebel, 1978, Interpolation Theory, Spaces of Besov–Hardy–Sobolev Type
Uspenskii, 1960, An imbedding theorem for S.L. Sobolevʼs classes Wpr of fractional order, Dokl. Akad. Nauk SSSR, 130, 992
Uspenskii, 1960, Properties of the classes Wpr with fractional derivatives on differentiable manifolds, Dokl. Akad. Nauk SSSR, 132, 60
Valdinoci, 2009, From the lung jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. SeMA, 49, 33
J.L. Vázquez, Nonlinear diffusion with fractional Laplacian operators, in: Proc. Abel Symp., in press, available online at http://www.uam.es/personal_pdi/ciencias/jvazquez/JLV-ABEL-2010.pdf.
Whitham, 1974, Linear and Nonlinear Waves
Zakharov, 1968, Stability of periodic waves of finite amplitude on the surface of a deep fluid, Zh. Prikl. Mekh. Tekh. Fiz., 9, 86
Ziemer, 1989
Y. Zhou, Fractional Sobolev extension and imbedding, preprint.