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December  2015, 35(12): 5631-5663. doi: 10.3934/dcds.2015.35.5631

Density estimates for vector minimizers and applications

Nicholas D. Alikakos 1, and  Giorgio Fusco 2,

1. 

Department of Mathematics, University of Athens, Panepistemiopolis, 15784 Athens, Greece
2. 

Università degli Studi dell'Aquila, Via Vetoio, 67010 Coppito, L'Aquila, Italy

Received  April 2014 Revised  September 2014 Published  May 2015

We extend the Caffarelli--Córdoba estimates to the vector case (L. Caffarelli and A. Córdoba, Uniform Convergence of a singular perturbation problem, Comm. Pure Appl. Math. 48 (1995)). In particular, we establish lower codimension density estimates. These are useful for studying the hierarchical structure of minimal solutions. We also give applications.
Keywords: polar form., Allen-Cahn functional, vector minimizers, structure of minimizers.
Mathematics Subject Classification: 35J20, 35J47, 35J5.
Citation: Nicholas D. Alikakos, Giorgio Fusco. Density estimates for vector minimizers and applications. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5631-5663. doi: 10.3934/dcds.2015.35.5631
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show all references

References:
[1]

Calc. Var. Part. Diff. Eqs., 5 (1997), 359-390. doi: 10.1007/s005260050071.  Google Scholar

[2]

Proc. Amer. Math. Soc., 139 (2011), 153-162. doi: 10.1090/S0002-9939-2010-10453-7.  Google Scholar

[3]

In Geometric Partial Differential Equations (eds. M. Novaga and G. Orlandi), Publications Scuola Normale Superiore, CRM Series, 15, Birkhäuser, 2013, 1-31. doi: 10.1007/978-88-7642-473-1_1.  Google Scholar

[4]

Comm. Partial Diff. Eqs, 37 (2012), 2093-2115. doi: 10.1080/03605302.2012.721851.  Google Scholar

[5]

Arch. Rat. Mech. Analysis, 202 (2011), 567-597. doi: 10.1007/s00205-011-0441-z.  Google Scholar

[6]

N. D. Alikakos and G. Fusco, In, preparation., ().   Google Scholar

[7]

Indiana Univ. Math. Journal, 57 (2008), 1871-1906. doi: 10.1512/iumj.2008.57.3181.  Google Scholar

[8]

Ann. Scuola Norm Sup. Pisa Cl. Sci., 9 (2009), 1-26.  Google Scholar

[9]

N. D. Alikakos and G. Fusco, A maximum principle for systems with variational structure and an application to standing waves,, to appear in JEMS, ().   Google Scholar

[10]

Ann. Inst. Henri Poincare, 7 (1990), 67-90.  Google Scholar

[11]

Birkhäuser, 1994. doi: 10.1007/978-1-4612-0287-5.  Google Scholar

[12]

Arch. Rat. Mech. Analysis, 124 (1993), 355-379. doi: 10.1007/BF00375607.  Google Scholar

[13]

Comm. Pure. Appl. Math., 49 (1996), 677-715. doi: 10.1002/(SICI)1097-0312(199607)49:7<677::AID-CPA2>3.0.CO;2-6.  Google Scholar

[14]

Comm. Pure Appl. Math., 48 (1995), 1-12. doi: 10.1002/cpa.3160480101.  Google Scholar

[15]

Journal Amer. Math. Society , 21 (2008), 847-862. doi: 10.1090/S0894-0347-08-00593-6.  Google Scholar

[16]

Archive for Rational Mechanics and Analysis, 216 (2015), 153-191. doi: 10.1007/s00205-014-0804-3.  Google Scholar

[17]

Graduate Studies in Mathematics, AMS, 1998. doi: 10.1090/gsm/019.  Google Scholar

[18]

Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.  Google Scholar

[19]

J. Funct. Anal., 214 (2004), 386-395. doi: 10.1016/j.jfa.2003.07.012.  Google Scholar

[20]

Calc. Var. Part. Diff. Eqs., 33 (2008), 1-35. doi: 10.1007/s00526-007-0146-1.  Google Scholar

[21]

Calculus of Variations and Partial Differential Equations, 47 (2013), 809-823. doi: 10.1007/s00526-012-0536-x.  Google Scholar

[22]

Calc. Var. Part. Diff. Eqs., 49 (2014), 963-985. doi: 10.1007/s00526-013-0607-7.  Google Scholar

[23]

Comm. Pure Appl. Anal., 13 (2014), 1045-1060. doi: 10.3934/cpaa.2014.13.1045.  Google Scholar

[24]

Trans. Amer. Math. Soc., 363 (2011), 4285-4307. doi: 10.1090/S0002-9947-2011-05356-0.  Google Scholar

[25]

Indiana Univ. Math. Journal, 32 (1983), 25-37. doi: 10.1512/iumj.1983.32.32003.  Google Scholar

[26]

Ind. Univ. Math. J., 57 (2008), 781-836. doi: 10.1512/iumj.2008.57.3089.  Google Scholar

[27]

SIAM J. Appl. Math., 49 (1989), 116-133. doi: 10.1137/0149007.  Google Scholar

[28]

SIAM J. Appl. Math., 49 (1989), 1722-1733. doi: 10.1137/0149104.  Google Scholar

[29]

Ann. of Math., 169 (2009), 41-78. doi: 10.4007/annals.2009.169.41.  Google Scholar

[30]

Journal de Mathématiques Pures et Appliquées, 101 (2014), 1-26. doi: 10.1016/j.matpur.2013.05.001.  Google Scholar

[31]

SIAM J. Math. Anal., 43 (2011), 2675-2687. doi: 10.1137/110831040.  Google Scholar

[32]

Interfaces And Free Boundaries, 14 (2012), 153-165. doi: 10.4171/IFB/277.  Google Scholar

[33]

P. Smyrnelis, Personal, communication., ().   Google Scholar

[34]

Rocky Mountain J. Math., 21 (1991), 799-807. doi: 10.1216/rmjm/1181072968.  Google Scholar

[35]

Ann. Math., 103 (1976), 489-539. doi: 10.2307/1970949.  Google Scholar

[36]

J. Reine Angew. Math., 574 (2004), 147-185. doi: 10.1515/crll.2004.068.  Google Scholar

[37]

Notes by O. Chodash., Stanford, 2012. Google Scholar

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