메뉴 건너뛰기




Volumn , Issue , 2009, Pages 74-96

The state of the art for a conjecture of de giorgi and related problems

Author keywords

[No Author keywords available]

Indexed keywords


EID: 84969706683     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1142/9789812834744_0004     Document Type: Chapter
Times cited : (73)

References (68)
  • 1
    • 0034957494 scopus 로고    scopus 로고
    • On a longstanding conjecture of e. De giorgi: Symmetry in 3d for general nonlinearities and a local minimality property
    • Antonio Avantaggiati on the occasion of his 70th birthday
    • Giovanni Alberti, Luigi Ambrosio, and Xavier Cabre. On a longstanding 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. Antonio Avantaggiati on the occasion of his 70th birthday.
    • (2001) Acta Appl. Math , vol.65 , Issue.1-3 , pp. 9-33
    • Alberti, G.1    Ambrosio, L.2    Cabre, X.3
  • 3
    • 0018480694 scopus 로고
    • A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening
    • S. Allen and J. Cahn. A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metallurgical 27:1084-1095, 1979.
    • (1979) Acta Metallurgical , vol.27 , pp. 1084-1095
    • Allen, S.1    Cahn, J.2
  • 4
    • 33845404914 scopus 로고
    • Existence and regularity for a minimum problem with free boundary
    • H. W. Alt and L. A. Caffarelli. Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math., 325:105-144, 1981.
    • (1981) J. Reine Angew. Math , vol.325 , pp. 105-144
    • Alt, H.W.1    Caffarelli, L.A.2
  • 5
    • 0347486747 scopus 로고    scopus 로고
    • Entire solutions of semilinear elliptic equations in r3 and a conjecture of de giorgi
    • 3 and a conjecture of De Giorgi. J. Amer. Math. Soc., 13(4):725-739 (electronic), 2000.
    • (2000) J. Amer. Math. Soc , vol.13 , Issue.4 , pp. 725-739
    • Ambrosio, L.1    Cabre, X.2
  • 6
    • 0034412523 scopus 로고    scopus 로고
    • The liouville property and a conjecture of de giorgi
    • Martin T. Barlow, Richard F. Bass, and Changfeng Gui. The Liouville property and a conjecture of De Giorgi. Comm. Pure Appl. Math., 53(8):1007-1038, 2000.
    • (2000) Comm. Pure Appl. Math , vol.53 , Issue.8 , pp. 1007-1038
    • Barlow, M.T.1    Bass, R.F.2    Gui, C.3
  • 7
    • 85015471238 scopus 로고    scopus 로고
    • Further qualitative properties for elliptic equations in unbounded domains
    • 1997. Dedicated to Ennio De Giorgi
    • Henri Berestycki, Luis Caffarelli, and Louis Nirenberg. Further qualitative properties for elliptic equations in unbounded domains. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 25(l-2):69-94 (1998), 1997. Dedicated to Ennio De Giorgi.
    • (1998) Ann. Scuola Norm. Sup. Pisa Cl. Sci , vol.25 , Issue.11-2 , pp. 69-94
    • Berestycki, H.1    Caffarelli, L.2    Nirenberg, L.3
  • 8
    • 0001327249 scopus 로고
    • Minimal cones and the bernstein problem
    • E. Bombieri, E. De Giorgi, and E. Giusti. 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
  • 9
    • 0003975247 scopus 로고    scopus 로고
    • Cambridge University Press, Cambridge
    • Jean Bertoin. Levy processes, volume 121 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1996.
    • (1996) Levy processes , vol.121
    • Bertoin, J.1
  • 10
    • 0001287814 scopus 로고    scopus 로고
    • One-dimensional symmetry of bounded entire solutions of some elliptic equations
    • Berestycki, F. Hamel, and R. Monneau. 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, F.H.1    Monneau, R.2
  • 12
    • 0346991527 scopus 로고    scopus 로고
    • A negative answer to a one-dimensional symmetry problem in the heisenberg group
    • H. Birindelli and E. Lanconelli. A negative answer to a one-dimensional symmetry problem in the Heisenberg group. Calc. Var. Partial Differential Equations, 18(4):357-372, 2003.
    • (2003) Calc. Var. Partial Differential Equations , vol.18 , Issue.4 , pp. 357-372
    • Birindelli, H.1    Lanconelli, E.2
  • 13
    • 0036345013 scopus 로고    scopus 로고
    • Monotonicity and symmetry results for degenerate elliptic equations on nilpotent lie groups
    • I. Birindelli and J. Prajapat. Monotonicity and symmetry results for degenerate elliptic equations on nilpotent Lie groups. Pacific J. Math., 204(1):1-17, 2002.
    • (2002) Pacific J. Math , vol.204 , Issue.1 , pp. 1-17
    • Birindelli, I.1    Prajapat, J.2
  • 14
    • 51449101940 scopus 로고    scopus 로고
    • The ginzburg-landau equation in the heisenberg group
    • Isabeau Birindelli and Enrico Valdinoci. The Ginzburg-Landau equation in the Heisenberg group. Commun. Contemp. Math., 10(5), 2008.
    • (2008) Commun. Contemp. Math , vol.10 , Issue.5
    • Birindelli, I.1    Valdinoci, E.2
  • 15
    • 85012254012 scopus 로고
    • Unicite et minimalite des solutions d’une equation de ginzburg-landau
    • Gilles Carbou. Unicite et minimalite des solutions d’une equation de Ginzburg-Landau. Ann. Inst. H. Poincare Anal. Non Lineaire, 12(3):305-318, 1995.
    • (1995) Ann. Inst. H. Poincare Anal. Non Lineaire , vol.12 , Issue.3 , pp. 305-318
    • Carbou, G.1
  • 17
    • 84990648271 scopus 로고
    • Uniform convergence of a singular perturbation problem
    • Luis A. Caffarelli and Antonio Cordoba. Uniform convergence of a singular perturbation problem. Comm,. Pure Appl. Math., 48(1):1-12, 1995.
    • (1995) Comm,. Pure Appl. Math , vol.48 , Issue.1 , pp. 1-12
    • Caffarelli, L.A.1    Cordoba, A.2
  • 18
    • 0028443419 scopus 로고
    • Hamiltonian long-wave approximations to the water-wave problem
    • Walter Craig and Mark D. Groves. 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
  • 19
    • 84990563948 scopus 로고
    • A gradient bound for entire solutions of quasi-linear equations and its consequences
    • Luis Caffarelli, Nicola Garofalo, and Fausto Segala. A gradient bound for entire solutions of quasi-linear equations and its consequences. Comm. Pure Appl. Math., 47(11):1457-1473, 1994.
    • (1994) Comm. Pure Appl. Math , vol.47 , Issue.11 , pp. 1457-1473
    • Caffarelli, L.1    Garofalo, N.2    Segala, F.3
  • 20
    • 0032264533 scopus 로고    scopus 로고
    • Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation
    • Diego Cordoba. Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation. Ann. of Math. (2), 148(3):1135-1152, 1998.
    • (1998) Ann. Of Math. , vol.148 , Issue.3 , pp. 1135-1152
    • Cordoba, D.1
  • 21
    • 34548348805 scopus 로고    scopus 로고
    • An extension problem related to the fractional laplacian
    • Luis Caffarelli and Luis Silvestre. An extension problem related to the fractional laplacian. Commun. in PDE, 32(8):1245, 2007.
    • (2007) Commun. In PDE , vol.32 , Issue.8 , pp. 1245
    • Caffarelli, L.1    Silvestre, L.2
  • 23
    • 30544441113 scopus 로고    scopus 로고
    • Layer solutions in a half-space for boundary reactions
    • Xavier Cabre and Joan Sola-Morales. Layer solutions in a half-space for boundary reactions. Comm. Pure Appl. Math., 58(12):1678-1732, 2005.
    • (2005) Comm. Pure Appl. Math , vol.58 , Issue.12 , pp. 1678-1732
    • Cabre, X.1    Sola-Morales, J.2
  • 25
    • 0000952106 scopus 로고
    • Saddle solutions of the bistable diffusion equation
    • Ha Dang, Paul C. Fife, and L. A. Peletier. Saddle solutions of the bistable diffusion equation. Z. Angew. Math. Phys., 43(6):984-998, 1992.
    • (1992) Z. Angew. Math. Phys , vol.43 , Issue.6 , pp. 984-998
    • Ha, D.1    Fife, P.C.2    Peletier, L.A.3
  • 27
    • 0036979160 scopus 로고    scopus 로고
    • Properties of entire solutions of non-uniformly elliptic equations arising in geometry and in phase transitions
    • Donatella Danielli and Nicola Garofalo. Properties of entire solutions of non-uniformly elliptic equations arising in geometry and in phase transitions. Calc. Var. Partial Differential Equations, 15(4):451-491, 2002.
    • (2002) Calc. Var. Partial Differential Equations , vol.15 , Issue.4 , pp. 451-491
    • Danielli, D.1    Garofalo, N.2
  • 29
    • 84969652460 scopus 로고    scopus 로고
    • Symmetry of global solutions to a class of fully nonlinear elliptic equations in 2d Preprint
    • D. De Silva and O. Savin. Symmetry of global solutions to a class of fully nonlinear elliptic equations in 2d. Preprint, 2008.
    • (2008)
    • De Silva, D.1    Savin, O.2
  • 30
    • 0001294525 scopus 로고    scopus 로고
    • Symmetry for solutions of semilinear elliptic equations in rn and related conjectures
    • Papers in memory of Ennio De Giorgi
    • N and related conjectures. Ricerche Mat., 48(suppl.):129- 154, 1999. Papers in memory of Ennio De Giorgi.
    • (1999) Ricerche Mat , vol.48
    • Farina, A.1
  • 32
  • 33
    • 84969684196 scopus 로고    scopus 로고
    • Liouville-type theorems for elliptic problems
    • Alberto Farina. Liouville-type theorems for elliptic problems. In
    • Farina, A.1
  • 34
    • 84969634333 scopus 로고    scopus 로고
    • Handbook of differential equations: Stationary partial differential equations Elsevier B. V., Amsterdam
    • M. Chipot, editor, Handbook of Differential Equations: Stationary Partial Differential Equations. Vol. 4, pages 61-116. Elsevier B. V., Amsterdam, 2007.
    • (2007) , vol.4-61 , Issue.116
    • Chipot, M.1
  • 35
    • 0002605789 scopus 로고
    • Relativistic stability of matter. I. Rev
    • C. Fefferman and R. de la Llave. Relativistic stability of matter. I. Rev. Mat. Iberoamericana, 2(1-2):119 213, 1986.
    • (1986) Mat. Iberoamericana , vol.2-119 , Issue.11-2 , pp. 213
    • Fefferman, C.1    De La Llave, R.2
  • 36
    • 57049094392 scopus 로고    scopus 로고
    • Bernstein and de giorgi type problems: New results via a geometric approach
    • (to appear)
    • Alberto Farina, Berardino Sciunzi, and Enrico Valdinoci. Bernstein and De Giorgi type problems: new results via a geometric approach. Ann. Scuola Norm. Sup. Pisa CL Sci. (to appear), 2008.
    • (2008) Ann. Scuola Norm. Sup. Pisa CL Sci
    • Farina, A.1    Sciunzi, B.2    Valdinoci, E.3
  • 37
    • 84969658833 scopus 로고    scopus 로고
    • Id symmetry for solutions of semilinear and quasilinear elliptic equations
    • Alberto Farina and Enrico Valdinoci. ID symmetry for solutions of semilinear and quasilinear elliptic equations. Preprint, 2008. http://www.math.ut exas.edu/mp_ar c.
    • (2008)
    • Farina, A.1    Valdinoci, E.2
  • 38
    • 38049018991 scopus 로고    scopus 로고
    • Geometry of quasiminimal phase transitions. Calc. Var
    • Alberto Farina and Enrico Valdinoci. Geometry of quasiminimal phase transitions. Calc. Var. Partial Differential Equations, 33(1):1- 35, 2008.
    • (2008) Partial Differential Equations , vol.33 , Issue.1
    • Farina, A.1    Valdinoci, E.2
  • 40
    • 0032373311 scopus 로고    scopus 로고
    • On a conjecture of de giorgi and some related problems
    • N. Ghoussoub and C. Gui. On a conjecture of De Giorgi and some related problems. Math. Ann., 311 (3):481-491, 1998.
    • (1998) Math. Ann , vol.311 , Issue.3 , pp. 481-491
    • Ghoussoub, N.1    Gui, C.2
  • 41
    • 0037226591 scopus 로고    scopus 로고
    • On de giorgi’s conjecture in dimensions 4 and 5
    • Nassif Ghoussoub and Changfeng Gui. On De Giorgi’s conjecture in dimensions 4 and 5. Ann. of Math. (2), 157(1):313—334, 2003.
    • (2003) Ann. Of Math , vol.157 , Issue.1 , pp. 313-334
    • Ghoussoub, N.1    Gui, C.2
  • 43
    • 0000187990 scopus 로고
    • On the theory of superfluidity
    • (1240-1245 Z. Eksper. Teoret. Fiz.)
    • V. L. Ginzburg and L. P. Pitaevskii. On the theory of superfluidity. Soviet Physics. JETP, 34 (7):858-861 (1240-1245 Z. Eksper. Teoret. Fiz.), 1958.
    • (1958) Soviet Physics. JETP , vol.34 , Issue.7 , pp. 858-861
    • Ginzburg, V.L.1    Pitaevskii, L.P.2
  • 44
    • 0001045117 scopus 로고    scopus 로고
    • Bogomol’nyi equation for intersecting domain walls
    • G. W. Gibbons and P. K. Townsend. Bogomol’nyi equation for intersecting domain walls. Phys. Rev. Lett., 83(9):1727-1730, Aug 1999.
    • (1999) Phys. Rev. Lett , vol.83 , Issue.9 , pp. 1727-1730
    • Gibbons, G.W.1    Townsend, P.K.2
  • 45
    • 0041625080 scopus 로고    scopus 로고
    • Elliptic partial differential equations of second order
    • Reprint of the 1998 edition
    • David Gilbarg and Neil S. Trudinger. Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.
    • (2001) Classics in Mathematics
    • Gilbarg, D.1    Trudinger, N.S.2
  • 46
    • 38049103351 scopus 로고    scopus 로고
    • Hamiltonian identities for elliptic partial differential equations
    • Changfeng Gui. Hamiltonian identities for elliptic partial differential equations. J. Fund. Anal., 254(4):904-933, 2008.
    • (2008) J. Fund. Anal , vol.254 , Issue.4 , pp. 904-933
    • Gui, C.1
  • 47
    • 15244346736 scopus 로고
    • Hypoelliptic second order differential equations
    • L. Hormander. Hypoelliptic second order differential equations. Acta Math., 119(1): 147—171, 1967.
    • (1967) Acta Math , vol.119 , Issue.1 , pp. 147-171
    • Hormander, L.1
  • 48
    • 33746362550 scopus 로고    scopus 로고
    • Towards a counter-example to a conjecture of de giorgi in high dimensions
    • David Jerison and Regis Monneau. Towards a counter-example to a conjecture of De Giorgi in high dimensions. Ann. Mat. Pura Appl. (A), 183(4):439-467, 2004.
    • (2004) Ann. Mat. Pura Appl. (A) , vol.183 , Issue.4 , pp. 439-467
    • Jerison, D.1    Monneau, R.2
  • 49
    • 0002859777 scopus 로고
    • New equations for the description of the motions of viscous incompressible fluids, and global solvability for their boundary value problems
    • O. A. Ladyzenskaja. New equations for the description of the motions of viscous incompressible fluids, and global solvability for their boundary value problems. Trudy Mat. Inst. Steklov., 102:85-104, 1967.
    • (1967) Trudy Mat. Inst. Steklov , vol.102 , pp. 85-104
    • Ladyzenskaja, O.A.1
  • 50
    • 84969745457 scopus 로고
    • Collected papers of l. D. Landau. Edited and with an introduction by d. Ter haar
    • Gordon and Breach Science Publishers, New York
    • L. D. Landau. Collected papers of L. D. Landau. Edited and with an introduction by D. ter Haar. Second printing. Gordon and Breach Science Publishers, New York, 1967.
    • (1967) Second Printing
    • Landau, L.D.1
  • 51
    • 0003643984 scopus 로고
    • Springer- Verlag, New York, Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften,, Band
    • N. S. Landkof. Foundations of modern potential theory. Springer- Verlag, New York, 1972. Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180.
    • (1972) Foundations of Modern Potential Theory , pp. 180
    • Landkof, N.S.1
  • 52
    • 0000359194 scopus 로고
    • Un esempio di p~-convergenza
    • Luciano Modica and Stefano Mortola. Un esempio di P~-convergenza. Boll. Un. Mat. Ital. 66 (5), 14(l):285-299, 1977.
    • (1977) Boll. Un. Mat. Ital , vol.14-66 , Issue.1-5 , pp. 285-299
    • Modica, L.1    Mortola, S.2
  • 53
    • 0000130264 scopus 로고
    • Some entire solutions in the plane of nonlinear poisson equations
    • Luciano Modica and Stefano Mortola. Some entire solutions in the plane of nonlinear Poisson equations. Boll. Un. Mat. Ital. B (5), 17(2):614-622, 1980.
    • (1980) Boll. Un. Mat. Ital. B (5) , vol.17 , Issue.2 , pp. 614-622
    • Modica, L.1    Mortola, S.2
  • 54
    • 84990623704 scopus 로고
    • And a liouville theorem for nonlinear poisson equations
    • Luciano Modica. A gradient bound and a Liouville theorem for nonlinear Poisson equations. Comm. Pure Appl. Math., 38(5):679-684, 1985.
    • (1985) Comm. Pure Appl. Math , vol.38 , Issue.5 , pp. 679-684
    • Modica, L.1    Bound, A.G.2
  • 55
    • 34250109116 scopus 로고
    • The gradient theory of phase transitions and the minimal interface criterion
    • Luciano Modica. The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal, 98(2):123-142, 1987.
    • (1987) Arch. Rational Mech. Anal , vol.98 , Issue.2 , pp. 123-142
    • Modica, L.1
  • 57
    • 84969680620 scopus 로고
    • Nonlinear nonlocal equations in the theory of waves Translated from the Russian manuscript by Boris Gommerstadt
    • P. I. Naumkin and I. A. Shishmarev. Nonlinear nonlocal equations in the theory of waves. American Mathematical Society, Providence, RI, 1994. Translated from the Russian manuscript by Boris Gommerstadt.
    • (1994)
    • Naumkin, P.I.1    Shishmarev, I.A.2
  • 59
    • 23744504356 scopus 로고    scopus 로고
    • thesis, University of Texas at Austin
    • Vasile Ovidiu Savin. Phase Transitions: Regularity of Flat Level Sets. PhD thesis, University of Texas at Austin, 2003. https://dspace.lib.utexas.edu/bitstream/2152/993/1/savinov032.pdf.
    • (2003) Phase Transitions: Regularity of Flat
    • Savin, V.O.1
  • 61
    • 21844482395 scopus 로고
    • On the stability of the saddle solution of allen-cahn’s equation
    • M. Schatzman. On the stability of the saddle solution of Allen-Cahn’s equation. Proc. Roy. Soc. Edinburgh Sect. A, 125(6):1241-1275, 1995.
    • (1995) Proc. Roy. Soc. Edinburgh Sect. A , vol.125 , Issue.6 , pp. 1241-1275
    • Schatzman, M.1
  • 62
    • 84969696672 scopus 로고
    • Singular integrals and differentiability properties of functions Princeton Mathematical Series
    • Elias M. Stein. Singular integrals and differentiability properties of functions. Princeton University Press, Princeton, N.J., 1970. Princeton Mathematical Series, No. 30.
    • (1970) , Issue.30
    • Stein, E.M.1
  • 63
    • 23744502733 scopus 로고    scopus 로고
    • Mean curvature properties for p-laplace phase transitions
    • Berardino Sciunzi and Enrico Valdinoci. Mean curvature properties for p-Laplace phase transitions. J. Eur. Math. Soc. (JEMS), 7(3):319-359, 2005.
    • (2005) J. Eur. Math. Soc. (JEMS) , vol.7 , Issue.3 , pp. 319-359
    • Sciunzi, B.1    Valdinoci, E.2
  • 65
    • 0032370858 scopus 로고    scopus 로고
    • Connectivity of phase boundaries in strictly convex domains
    • Peter Sternberg and Kevin Zumbrun. Connectivity of phase boundaries in strictly convex domains. Arch. Rational Mech. Anal., 141(4):375-400, 1998.
    • (1998) Arch. Rational Mech. Anal , vol.141 , Issue.4 , pp. 375-400
    • Sternberg, P.1    Zumbrun, K.2
  • 66
    • 0032221205 scopus 로고    scopus 로고
    • A poincare inequality with applications to volume-constrained area-minimizing surfaces
    • Peter Sternberg and Kevin Zumbrun. A Poincare inequality with applications to volume-constrained area-minimizing surfaces. J. Reine Angew. Math., 503:63-85, 1998.
    • (1998) J. Reine Angew. Math , vol.503 , pp. 63-85
    • Sternberg, P.1    Zumbrun, K.2
  • 67
    • 37849185621 scopus 로고    scopus 로고
    • Flatness of bernoulli jets
    • Enrico Valdinoci. Flatness of Bernoulli jets. Math. Z., 254(2):257-298, 2006.
    • (2006) Math. Z , vol.254 , Issue.2 , pp. 257-298
    • Valdinoci, E.1
  • 68
    • 33646346403 scopus 로고    scopus 로고
    • Flat level set regularity of p-laplace phase transitions
    • Enrico Valdinoci, Berardino Sciunzi, and Vasile Ovidiu Savin. Flat level set regularity of p-Laplace phase transitions. Mem. Amer. Math. Soc., 182(858):148, 2006.
    • (2006) Mem. Amer. Math. Soc , vol.182 , Issue.858 , pp. 148
    • Valdinoci, E.1    Sciunzi, B.2    Savin, V.O.3


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.