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Volumn 2, Issue , 2013, Pages 117-130

Geometric analysis of fractional phase transition interfaces

Author keywords

Asymptotics; Fractional Laplacian; Nonlocal minimal surfaces; Rigidity and regularity theory

Indexed keywords


EID: 84940423705     PISSN: 2281518X     EISSN: 22815198     Source Type: Book Series    
DOI: 10.1007/978-88-470-2841-8_8     Document Type: Article
Times cited : (9)

References (62)
  • 1
    • 0032383890 scopus 로고    scopus 로고
    • A nonlocal anisotropic model for phase transitions, I. The optimal profile problem
    • Alberti, G., Bellettini, G.: A nonlocal anisotropic model for phase transitions, I. The optimal profile problem. Math. Ann. 310(3), 527–560 (1998)
    • (1998) Math. Ann. , vol.310 , Issue.3 , pp. 527-560
    • Alberti, G.1    Bellettini, G.2
  • 4
    • 0034957494 scopus 로고    scopus 로고
    • On a long-standing conjecture of E. De Giorgi: Symmetry in 3D for general nonlinearities and a local minimality property
    • Alberti, G., Ambrosio, L., Cabré, X.: On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property. Acta Appl. Math. 65(1–3), 9–33 (2001)
    • (2001) Acta Appl. Math , vol.65 , Issue.13 , pp. 9-33
    • Alberti, G.1    Ambrosio, L.2    Cabré, X.3
  • 5
    • 0347486747 scopus 로고    scopus 로고
    • Entire solutions of semilinear elliptic equations in R3 and a conjecture of De Giorgi
    • Ambrosio, L., Cabré, X.: Entire solutions of semilinear elliptic equations in R3 and a conjecture of De Giorgi. J. Am. Math. Soc. 13, 725–739 (2000)
    • (2000) J. Am. Math. Soc , vol.13 , pp. 725-739
    • Ambrosio, L.1    Cabré, X.2
  • 6
    • 78751574089 scopus 로고    scopus 로고
    • Γ -convergence of nonlocal perimeter functionals
    • Ambrosio, L., de Philippis, G., Martinazzi, L.: Γ -convergence of nonlocal perimeter functionals. Manuscr. Math. 134, 377–403 (2011)
    • (2011) Manuscr. Math , vol.134 , pp. 377-403
    • Ambrosio, L.1    de Philippis, G.2    Martinazzi, L.3
  • 7
    • 0034412523 scopus 로고    scopus 로고
    • The Liouville property and a conjecture of De Giorgi
    • Barlow, M.T., Bass, R.F., Gui, C.: The Liouville property and a conjecture of De Giorgi. Commun. Pure Appl. Math. 53(8), 1007–1038 (2000)
    • (2000) Commun. Pure Appl. Math. , vol.53 , Issue.8 , pp. 1007-1038
    • Barlow, M.T.1    Bass, R.F.2    Gui, C.3
  • 8
    • 84875607189 scopus 로고    scopus 로고
    • Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces
    • accepted for publication
    • Barrios Barrera, B., Figalli, A., Valdinoci, E.: Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (2012, accepted for publication)
    • (2012) Ann. Sc. Norm. Super. Pisa, Cl. Sci.
    • Barrios Barrera, B.1    Figalli, A.2    Valdinoci, E.3
  • 9
    • 85015471238 scopus 로고    scopus 로고
    • Further qualitative properties for elliptic equations in unbounded domains
    • (Dedicated to Ennio De Giorgi)
    • Berestycki, H., Caffarelli, L., Nirenberg, L.: Further qualitative properties for elliptic equations in unbounded domains. Ann. Sc. Norm. Super. Pisa. Cl. Sci. (4) 25(1–2), 69–94 (1997–1998) (Dedicated to Ennio De Giorgi)
    • (1997) Ann. Sc. Norm. Super. Pisa. Cl. Sci. (4) , vol.25 , Issue.12 , pp. 69-94
    • Berestycki, H.1    Caffarelli, L.2    Nirenberg, L.3
  • 10
    • 0001287814 scopus 로고    scopus 로고
    • One-dimensional symmetry of bounded entire solutions of some elliptic equations
    • Berestycki, H., Hamel, F., Monneau, R.: One-dimensional symmetry of bounded entire solutions of some elliptic equations. Duke Math. J. 103(3), 375–396 (2000)
    • (2000) Duke Math. J. , vol.103 , Issue.3 , pp. 375-396
    • Berestycki, H.1    Hamel, F.2    Monneau, R.3
  • 11
    • 0001327249 scopus 로고
    • Minimal cones and the Bernstein problem
    • Bombieri, E., De Giorgi, E., Giusti, E.: Minimal cones and the Bernstein problem. Invent. Math. 7, 243–268 (1969)
    • (1969) Invent. Math. , vol.7 , pp. 243-268
    • Bombieri, E.1    De Giorgi, E.2    Giusti, E.3
  • 12
    • 0141580732 scopus 로고    scopus 로고
    • How to recognize constant functions. Connections with Sobolev spaces
    • Brezis, H.: How to recognize constant functions. Connections with Sobolev spaces. Usp. Mat. Nauk 57(346), 59–74 (2002)
    • (2002) Usp. Mat. Nauk , vol.57 , Issue.346 , pp. 59-74
    • Brezis, H.1
  • 13
    • 77955326615 scopus 로고    scopus 로고
    • Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian
    • Cabré, X., Cinti, E.: Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian. Discrete Contin. Dyn. Syst. 28(3), 1179–1206 (2010)
    • (2010) Discrete Contin. Dyn. Syst. , vol.28 , Issue.3 , pp. 1179-1206
    • Cabré, X.1    Cinti, E.2
  • 14
    • 85035126472 scopus 로고    scopus 로고
    • Sharp energy estimates for nonlinear fractional diffusion equations
    • accepted for publication
    • Cabré, X., Cinti, E.: Sharp energy estimates for nonlinear fractional diffusion equations. Calc. Var. Partial Differ. Equ. (2012, accepted for publication)
    • (2012) Calc. Var. Partial Differ. Equ.
    • Cabré, X.1    Cinti, E.2
  • 16
    • 30544441113 scopus 로고    scopus 로고
    • Layer solutions in a half-space for boundary reactions
    • Cabré, X., Solà-Morales, J.: Layer solutions in a half-space for boundary reactions. Commun. Pure Appl. Math. 58(12), 1678–1732 (2005)
    • (2005) Commun. Pure Appl. Math. , vol.58 , Issue.12 , pp. 1678-1732
    • Cabré, X.1    Solà-Morales, J.2
  • 17
    • 84990648271 scopus 로고
    • Uniform convergence of a singular perturbation problem
    • Caffarelli, L., Córdoba, A.: Uniform convergence of a singular perturbation problem. Commun. Pure Appl. Math. 48, 1–12 (1995)
    • (1995) Commun. Pure Appl. Math , vol.48 , pp. 1-12
    • Caffarelli, L.1    Córdoba, A.2
  • 18
    • 79952985705 scopus 로고    scopus 로고
    • Uniform estimates and limiting arguments for nonlocal minimal surfaces
    • Caffarelli, L., Valdinoci, E.: Uniform estimates and limiting arguments for nonlocal minimal surfaces. Calc. Var. Partial Differ. Equ. 41(1–2), 203–240 (2011)
    • (2011) Calc. Var. Partial Differ. Equ , vol.41 , Issue.12 , pp. 203-240
    • Caffarelli, L.1    Valdinoci, E.2
  • 20
    • 37849049246 scopus 로고    scopus 로고
    • Regularity estimates for the solution and the free boundary of the obstacle problemfor the fractional Laplacian
    • Caffarelli, L., Salsa, S., Silvestre, L.: Regularity estimates for the solution and the free boundary of the obstacle problemfor the fractional Laplacian. Invent. Math. 171(2), 425–461 (2008)
    • (2008) Invent. Math. , vol.171 , Issue.2 , pp. 425-461
    • Caffarelli, L.1    Salsa, S.2    Silvestre, L.3
  • 22
    • 77955505190 scopus 로고    scopus 로고
    • Variational problems for free boundaries for the fractional Laplacian
    • Caffarelli, L., Roquejoffre, J.M., Sire, Y.: Variational problems for free boundaries for the fractional Laplacian. J. Eur. Math. Soc. 12(5), 1151–1179 (2010)
    • (2010) J. Eur. Math. Soc , vol.12 , Issue.5 , pp. 1151-1179
    • Caffarelli, L.1    Roquejoffre, J.M.2    Sire, Y.3
  • 24
    • 0028443419 scopus 로고
    • Hamiltonian long-wave approximations to the water-wave problem
    • Craig, W., Groves, M.D.: Hamiltonian long-wave approximations to the water-wave problem. Wave Motion 19(4), 367–389 (1994)
    • (1994) Wave Motion , vol.19 , Issue.4 , pp. 367-389
    • Craig, W.1    Groves, M.D.2
  • 25
    • 0001654990 scopus 로고    scopus 로고
    • The modulational regime of three-dimensional water waves and the Davey-Stewartson system
    • Craig, W., Schanz, U., Sulem, C.: The modulational regime of three-dimensional water waves and the Davey-Stewartson system. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 5(14), 615– 667 (1997)
    • (1997) Ann. Inst. Henri Poincaré, Anal. Non Linéaire , vol.5 , Issue.14 , pp. 615-667
    • Craig, W.1    Schanz, U.2    Sulem, C.3
  • 27
    • 57049180714 scopus 로고    scopus 로고
    • A counterexample to a conjecture by De Giorgi in large dimensions
    • del Pino, M., Kowalczyk, M., Wei, J.: A counterexample to a conjecture by De Giorgi in large dimensions. C. R. Math. 346, 23–24 (2008)
    • (2008) C. R. Math. , vol.346 , pp. 23-24
    • Del Pino, M.1    Kowalczyk, M.2    Wei, J.3
  • 28
    • 84878388384 scopus 로고    scopus 로고
    • Entire solutions of the Allen-Cahn equation and complete embedded minimal surfaces
    • del Pino, M., Kowalczyk, M., Wei, J.: Entire solutions of the Allen-Cahn equation and complete embedded minimal surfaces. Rev. Unión Mat. Argent. 50(2), 95–107 (2009)
    • (2009) Rev. Unión Mat. Argent. , vol.50 , Issue.2 , pp. 95-107
    • Del Pino, M.1    Kowalczyk, M.2    Wei, J.3
  • 29
    • 80054998436 scopus 로고    scopus 로고
    • On De Giorgi’s conjecture in dimension N ≥ 9
    • del Pino, M., Kowalczyk, M., Wei, J.: On De Giorgi’s conjecture in dimension N ≥ 9. Ann. Math. 174, 1485–1569 (2011)
    • (2011) Ann. Math. , vol.174 , pp. 1485-1569
    • Del Pino, M.1    Kowalczyk, M.2    Wei, J.3
  • 30
    • 84863469913 scopus 로고    scopus 로고
    • Hitchhiker.S guide to the fractional Sobolev spaces
    • Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker.s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012). doi:10.1016/j.bulsci.2011.12.004. http://www. sciencedirect.com/science/article/pii/S0007449711001254
    • (2012) Bull. Sci. Math , vol.136 , Issue.5 , pp. 521-573
    • Di Nezza, E.1    Palatucci, G.2    Valdinoci, E.3
  • 32
    • 0001294525 scopus 로고    scopus 로고
    • N and related conjectures
    • N and related conjectures. Ric. Mat. 48, 129–154 (1999)
    • (1999) Ric. Mat. , vol.48 , pp. 129-154
    • Farina, A.1
  • 33
    • 80051924152 scopus 로고    scopus 로고
    • The state of art for a conjecture of De Giorgi and related questions
    • World Scientific, Singapore
    • Farina, A., Valdinoci, E.: The state of art for a conjecture of De Giorgi and related questions. In: Reaction-Diffusion Systems and Viscosity Solutions. World Scientific, Singapore (2008)
    • (2008) Reaction-Diffusion Systems and Viscosity Solutions
    • Farina, A.1    Valdinoci, E.2
  • 34
    • 84859732880 scopus 로고    scopus 로고
    • Rigidity results for elliptic PDEs with uniform limits: An abstract framework with applications
    • Farina, A., Valdinoci, E.: Rigidity results for elliptic PDEs with uniform limits: an abstract framework with applications. Indiana Univ. Math. J. 60, 121–142 (2011)
    • (2011) Indiana Univ. Math. J. , vol.60 , pp. 121-142
    • Farina, A.1    Valdinoci, E.2
  • 35
    • 84966226124 scopus 로고
    • The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension
    • Federer, H.: The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension. Bull. Am. Math. Soc. 76, 767–771 (1970)
    • (1970) Bull. Am. Math. Soc. , vol.76 , pp. 767-771
    • Federer, H.1
  • 36
  • 37
    • 27844442719 scopus 로고    scopus 로고
    • Γ -limit of a phase-field model of dislocations
    • Garroni, A., Müller, S.: Γ -limit of a phase-field model of dislocations. SIAM J. Math. Anal. 36, 1943–1964 (2005)
    • (2005) SIAM J. Math. Anal , vol.36 , pp. 1943-1964
    • Garroni, A.1    Müller, S.2
  • 38
    • 33745603193 scopus 로고    scopus 로고
    • A variational model for dislocations in the line tension limit
    • Garroni, A., Müller, S.: A variational model for dislocations in the line tension limit. Arch. Ration. Mech. Anal. 181, 535–578 (2006)
    • (2006) Arch. Ration. Mech. Anal. , vol.181 , pp. 535-578
    • Garroni, A.1    Müller, S.2
  • 40
    • 0032373311 scopus 로고    scopus 로고
    • On a conjecture of De Giorgi and some related problems
    • Ghoussoub, N., Gui, C.: On a conjecture of De Giorgi and some related problems. Math. Ann. 311, 481–491 (1998)
    • (1998) Math. Ann. , vol.311 , pp. 481-491
    • Ghoussoub, N.1    Gui, C.2
  • 41
    • 70350180632 scopus 로고    scopus 로고
    • Γ -convergence of an energy functional related to the fractional Laplacian
    • González, M.D.M.: Γ -convergence of an energy functional related to the fractional Laplacian. Calc. Var. Partial Differ. Equ. 36, 173–210 (2009)
    • (2009) Calc. Var. Partial Differ. Equ , vol.36 , pp. 173-210
    • González, M.D.M.1
  • 42
    • 33645787821 scopus 로고    scopus 로고
    • A nonlocal singular perturbation problem with periodic well potential
    • (electronic)
    • Kurzke, M.: A nonlocal singular perturbation problem with periodic well potential. ESAIM Control Optim. Calc. Var. 12(1), 52–63 (2006) (electronic)
    • (2006) ESAIM Control Optim. Calc. Var. , vol.12 , Issue.1 , pp. 52-63
    • Kurzke, M.1
  • 43
    • 0000669501 scopus 로고    scopus 로고
    • A 2D model for quasigeostrophic flow: Comparison with the two-dimensional Euler flow
    • Nonlinear phenomena in ocean dynamics, Los Alamos, NM (1995)
    • Majda, A.J., Tabak, E.G.: A 2D model for quasigeostrophic flow: comparison with the two-dimensional Euler flow. Physica D 98(2–4), 515–522 (1996). Nonlinear phenomena in ocean dynamics, Los Alamos, NM (1995)
    • (1996) Physica D , vol.98 , Issue.24 , pp. 515-522
    • Majda, A.J.1    Tabak, E.G.2
  • 44
    • 0037058485 scopus 로고    scopus 로고
    • On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces
    • Maz’ya, V., Shaposhnikova, T.: On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces. J. Funct. Anal. 195(2), 230–238 (2002)
    • (2002) J. Funct. Anal. , vol.195 , Issue.2 , pp. 230-238
    • Maz’Ya, V.1    Shaposhnikova, T.2
  • 47
    • 34250109116 scopus 로고
    • The gradient theory of phase transitions and the minimal interface criterion
    • Modica, L.: The gradient theory of phase transitions and the minimal interface criterion. Arch. Ration. Mech. Anal. 98, 23–142 (1987)
    • (1987) Arch. Ration. Mech. Anal , vol.98 , pp. 23-142
    • Modica, L.1
  • 49
    • 84855181783 scopus 로고    scopus 로고
    • Local and global minimizers for a variational energy involving a fractional norm
    • Palatucci, G., Savin, O., Valdinoci, E.: Local and global minimizers for a variational energy involving a fractional norm. Ann. Mat. Pura Appl. (2012). doi:10.1007/s10231-011-0243-9
    • (2012) Ann. Mat. Pura Appl.
    • Palatucci, G.1    Savin, O.2    Valdinoci, E.3
  • 50
    • 67249165287 scopus 로고    scopus 로고
    • Regularity of flat level sets in phase transitions
    • Savin, O.: Regularity of flat level sets in phase transitions. Ann. Math. 169(1), 41–78 (2009)
    • (2009) Ann. Math. , vol.169 , Issue.1 , pp. 41-78
    • Savin, O.1
  • 51
    • 84867924384 scopus 로고    scopus 로고
    • Elliptic PDEs with fibered nonlinearities
    • Savin, O., Valdinoci, E.: Elliptic PDEs with fibered nonlinearities. J. Geom. Anal. 19(2), 420– 432 (2009)
    • (2009) J. Geom. Anal. , vol.19 , Issue.2 , pp. 420-432
    • Savin, O.1    Valdinoci, E.2
  • 52
    • 84856287508 scopus 로고    scopus 로고
    • Density estimates for a nonlocal variational model via the Sobolev inequality
    • Savin, O., Valdinoci, E.: Density estimates for a nonlocal variational model via the Sobolev inequality. SIAM J. Math. Anal. 43, 2675–2687 (2011)
    • (2011) SIAM J. Math. Anal. , vol.43 , pp. 2675-2687
    • Savin, O.1    Valdinoci, E.2
  • 56
    • 2542448738 scopus 로고
    • Questioni di elasticità non linearizzata e semilinearizzata
    • Signorini, A.: Questioni di elasticità non linearizzata e semilinearizzata. Rend. Mat. Appl. 18, 95–139 (1959). http://www.scienzadellecostruzioni.co.uk/Documenti/Signorini-Prima.pdf
    • (1959) Rend. Mat. Appl. , vol.18 , pp. 95-139
    • Signorini, A.1
  • 57
    • 33751514593 scopus 로고    scopus 로고
    • Regularity of the obstacle problem for a fractional power of the Laplace operator
    • Silvestre, L.: Regularity of the obstacle problem for a fractional power of the Laplace operator. Commun. Pure Appl. Math. 60(1), 67–112 (2007)
    • (2007) Commun. Pure Appl. Math. , vol.60 , Issue.1 , pp. 67-112
    • Silvestre, L.1
  • 58
    • 59849109073 scopus 로고    scopus 로고
    • Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result
    • Sire, Y., Valdinoci, E.: Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result. J. Funct. Anal. 256(6), 1842–1864 (2009)
    • (2009) J. Funct. Anal. , vol.256 , Issue.6 , pp. 1842-1864
    • Sire, Y.1    Valdinoci, E.2
  • 59
    • 0031116613 scopus 로고    scopus 로고
    • The Peierls-Nabarro and Benjamin-Ono equations
    • Toland, J.F.: The Peierls-Nabarro and Benjamin-Ono equations. J. Funct. Anal. 145(1), 136– 150 (1997)
    • (1997) J. Funct. Anal. , vol.145 , Issue.1 , pp. 136-150
    • Toland, J.F.1
  • 60
    • 80052309087 scopus 로고    scopus 로고
    • From the long jump random walk to the fractional Laplacian
    • Valdinoci, E.: From the long jump random walk to the fractional Laplacian. Bol. Soc. Esp. Mat. Apl. SeMA 49, 33–34 (2009)
    • (2009) Bol. Soc. Esp. Mat. Apl. Sema , vol.49 , pp. 33-34
    • Valdinoci, E.1
  • 61
    • 33748354629 scopus 로고
    • Generalized coarea formula and fractal sets
    • Visintin, A.: Generalized coarea formula and fractal sets. Jpn. J. Ind. Appl. Math. 8, 175–201 (1991)
    • (1991) Jpn. J. Ind. Appl. Math. , vol.8 , pp. 175-201
    • Visintin, A.1


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