# American Institute of Mathematical Sciences

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

## Density estimates for vector minimizers and applications

 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.
Citation: Nicholas D. Alikakos, Giorgio Fusco. Density estimates for vector minimizers and applications. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5631-5663. doi: 10.3934/dcds.2015.35.5631
##### References:
 [1] S. Alama, L. Bronsard and C. Gui, Stationary solutions in $\mathbbR^2$ for an Allen-Cahn system with multiple well potential,, Calc. Var. Part. Diff. Eqs., 5 (1997), 359. doi: 10.1007/s005260050071. Google Scholar [2] N. D. Alikakos, Some basic facts on the system $\Delta u-W_u(u)=0$,, Proc. Amer. Math. Soc., 139 (2011), 153. doi: 10.1090/S0002-9939-2010-10453-7. Google Scholar [3] N. D. Alikakos, On the structure of phase transition maps for three or more coexisting phases,, In Geometric Partial Differential Equations (eds. M. Novaga and G. Orlandi), (2013), 1. doi: 10.1007/978-88-7642-473-1_1. Google Scholar [4] N. D. Alikakos, A new proof for the existence of an equivariant entire solution connecting the minima of the potential for the system $\Delta u-W_u(u)=0$,, Comm. Partial Diff. Eqs, 37 (2012), 2093. doi: 10.1080/03605302.2012.721851. Google Scholar [5] N. D. Alikakos and G. Fusco, Entire solutions to equivariant elliptic systems with variational structure,, Arch. Rat. Mech. Analysis, 202 (2011), 567. doi: 10.1007/s00205-011-0441-z. Google Scholar [6] N. D. Alikakos and G. Fusco, In, preparation., (). Google Scholar [7] N. D. Alikakos and G. Fusco, On the connection problem for potentials with several global minima,, Indiana Univ. Math. Journal, 57 (2008), 1871. doi: 10.1512/iumj.2008.57.3181. Google Scholar [8] N. D. Alikakos and G. Fusco, Asymptotic and rigidity results for symmetric solutions of the elliptic system $\Delta u=W_u(u)$,, Ann. Scuola Norm Sup. Pisa Cl. Sci., 9 (2009), 1. 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] S. Baldo, Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids,, Ann. Inst. Henri Poincare, 7 (1990), 67. Google Scholar [11] F. Bethuel, H. Brezis and F. Helein, Ginzburg-Landau Vortices,, Birkhäuser, (1994). doi: 10.1007/978-1-4612-0287-5. Google Scholar [12] L. Bronsard and F. Reitich, On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation,, Arch. Rat. Mech. Analysis, 124 (1993), 355. doi: 10.1007/BF00375607. Google Scholar [13] L. Bronsard, C. Gui and M. Schatzman, A three-layered minimizer in $\mathbbR^2$ for a variational problem with a symmetric three-well potential,, Comm. Pure. Appl. Math., 49 (1996), 677. doi: 10.1002/(SICI)1097-0312(199607)49:7<677::AID-CPA2>3.0.CO;2-6. Google Scholar [14] L. Caffarelli and A. Cordoba, Uniform convergence of a singular perturbation problem,, Comm. Pure Appl. Math., 48 (1995), 1. doi: 10.1002/cpa.3160480101. Google Scholar [15] L. Caffarelli and F. Lin, Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries,, Journal Amer. Math. Society , 21 (2008), 847. doi: 10.1090/S0894-0347-08-00593-6. Google Scholar [16] A. Cesaroni, C. M. Muratov and M. Novaga, Front propagation and phase field models of stratified media,, Archive for Rational Mechanics and Analysis, 216 (2015), 153. doi: 10.1007/s00205-014-0804-3. Google Scholar [17] L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998). doi: 10.1090/gsm/019. Google Scholar [18] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, Studies in Advanced Mathematics, (1992). Google Scholar [19] A. Farina, Two results on entire solutions of Ginzburg-Landau systems in higher dimensions,, J. Funct. Anal., 214 (2004), 386. doi: 10.1016/j.jfa.2003.07.012. Google Scholar [20] A. Farina and E. Valdinoci, Geometry of quasiminimal phase transitions,, Calc. Var. Part. Diff. Eqs., 33 (2008), 1. doi: 10.1007/s00526-007-0146-1. Google Scholar [21] M. Fazly and N. Ghoussoub, De Giorgi type results for elliptic systems,, Calculus of Variations and Partial Differential Equations, 47 (2013), 809. doi: 10.1007/s00526-012-0536-x. Google Scholar [22] G. Fusco, Equivariant entire solutions to the elliptic system $\Delta u=W_u(u)$ for general $G-$invariant potentials,, Calc. Var. Part. Diff. Eqs., 49 (2014), 963. doi: 10.1007/s00526-013-0607-7. Google Scholar [23] G. Fusco, On some elementary properties of vector minimizers of the Allen-Cahn energy,, Comm. Pure Appl. Anal., 13 (2014), 1045. doi: 10.3934/cpaa.2014.13.1045. Google Scholar [24] G. Fusco, F. Leonetti and C. Pignotti, A uniform estimate for positive solutions of semilinear elliptic equations,, Trans. Amer. Math. Soc., 363 (2011), 4285. doi: 10.1090/S0002-9947-2011-05356-0. Google Scholar [25] E. Gonzalez, U. Massari and I. Tamanini, On the regularity of boundaries of sets minimizing perimeter with a volume constraint,, Indiana Univ. Math. Journal, 32 (1983), 25. doi: 10.1512/iumj.1983.32.32003. Google Scholar [26] C. Gui and M. Schatzman, Symmetric quadruple phase transitions,, Ind. Univ. Math. J., 57 (2008), 781. doi: 10.1512/iumj.2008.57.3089. Google Scholar [27] J. Rubinstein, P. Sternberg and J. Keller, Fast reaction, slow diffusion and curve shortening,, SIAM J. Appl. Math., 49 (1989), 116. doi: 10.1137/0149007. Google Scholar [28] J. Rubinstein, P. Sternberg and J. Keller, Reaction-Diffusion processes and evolution to harmonic maps,, SIAM J. Appl. Math., 49 (1989), 1722. doi: 10.1137/0149104. Google Scholar [29] O. Savin, Regularity of flat level sets in phase transitions,, Ann. of Math., 169 (2009), 41. doi: 10.4007/annals.2009.169.41. Google Scholar [30] O. Savin and E. Valdinoci, Density estimates for a variational model driven by the Gagliardo norm,, Journal de Mathématiques Pures et Appliquées, 101 (2014), 1. doi: 10.1016/j.matpur.2013.05.001. Google Scholar [31] O. Savin and E. Valdinoci, Density estimates for a nonolocal variational model via the Sobolev inequality,, SIAM J. Math. Anal., 43 (2011), 2675. doi: 10.1137/110831040. Google Scholar [32] Y. Sire and E. Valdinoci, Density estimates for phase transitions with a trace,, Interfaces And Free Boundaries, 14 (2012), 153. doi: 10.4171/IFB/277. Google Scholar [33] P. Smyrnelis, Personal, communication., (). Google Scholar [34] P. Sternberg, Vector-valued local minimizers of nonconvex variational problems,, Rocky Mountain J. Math., 21 (1991), 799. doi: 10.1216/rmjm/1181072968. Google Scholar [35] J. E. Taylor, The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces,, Ann. Math., 103 (1976), 489. doi: 10.2307/1970949. Google Scholar [36] E. Valdinoci, Plane-like minimizers in periodic media: Jet flows and Ginzburg-Landau-type functionals,, J. Reine Angew. Math., 574 (2004), 147. doi: 10.1515/crll.2004.068. Google Scholar [37] B. White, Topics in GMT,, Notes by O. Chodash., (2012). Google Scholar

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##### References:
 [1] S. Alama, L. Bronsard and C. Gui, Stationary solutions in $\mathbbR^2$ for an Allen-Cahn system with multiple well potential,, Calc. Var. Part. Diff. Eqs., 5 (1997), 359. doi: 10.1007/s005260050071. Google Scholar [2] N. D. Alikakos, Some basic facts on the system $\Delta u-W_u(u)=0$,, Proc. Amer. Math. Soc., 139 (2011), 153. doi: 10.1090/S0002-9939-2010-10453-7. Google Scholar [3] N. D. Alikakos, On the structure of phase transition maps for three or more coexisting phases,, In Geometric Partial Differential Equations (eds. M. Novaga and G. Orlandi), (2013), 1. doi: 10.1007/978-88-7642-473-1_1. Google Scholar [4] N. D. Alikakos, A new proof for the existence of an equivariant entire solution connecting the minima of the potential for the system $\Delta u-W_u(u)=0$,, Comm. Partial Diff. Eqs, 37 (2012), 2093. doi: 10.1080/03605302.2012.721851. Google Scholar [5] N. D. Alikakos and G. Fusco, Entire solutions to equivariant elliptic systems with variational structure,, Arch. Rat. Mech. Analysis, 202 (2011), 567. doi: 10.1007/s00205-011-0441-z. Google Scholar [6] N. D. Alikakos and G. Fusco, In, preparation., (). Google Scholar [7] N. D. Alikakos and G. Fusco, On the connection problem for potentials with several global minima,, Indiana Univ. Math. Journal, 57 (2008), 1871. doi: 10.1512/iumj.2008.57.3181. Google Scholar [8] N. D. Alikakos and G. Fusco, Asymptotic and rigidity results for symmetric solutions of the elliptic system $\Delta u=W_u(u)$,, Ann. Scuola Norm Sup. Pisa Cl. Sci., 9 (2009), 1. 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] S. Baldo, Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids,, Ann. Inst. Henri Poincare, 7 (1990), 67. Google Scholar [11] F. Bethuel, H. Brezis and F. Helein, Ginzburg-Landau Vortices,, Birkhäuser, (1994). doi: 10.1007/978-1-4612-0287-5. Google Scholar [12] L. Bronsard and F. Reitich, On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation,, Arch. Rat. Mech. Analysis, 124 (1993), 355. doi: 10.1007/BF00375607. Google Scholar [13] L. Bronsard, C. Gui and M. Schatzman, A three-layered minimizer in $\mathbbR^2$ for a variational problem with a symmetric three-well potential,, Comm. Pure. Appl. Math., 49 (1996), 677. doi: 10.1002/(SICI)1097-0312(199607)49:7<677::AID-CPA2>3.0.CO;2-6. Google Scholar [14] L. Caffarelli and A. Cordoba, Uniform convergence of a singular perturbation problem,, Comm. Pure Appl. Math., 48 (1995), 1. doi: 10.1002/cpa.3160480101. Google Scholar [15] L. Caffarelli and F. Lin, Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries,, Journal Amer. Math. Society , 21 (2008), 847. doi: 10.1090/S0894-0347-08-00593-6. Google Scholar [16] A. Cesaroni, C. M. Muratov and M. Novaga, Front propagation and phase field models of stratified media,, Archive for Rational Mechanics and Analysis, 216 (2015), 153. doi: 10.1007/s00205-014-0804-3. Google Scholar [17] L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998). doi: 10.1090/gsm/019. Google Scholar [18] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, Studies in Advanced Mathematics, (1992). Google Scholar [19] A. Farina, Two results on entire solutions of Ginzburg-Landau systems in higher dimensions,, J. Funct. Anal., 214 (2004), 386. doi: 10.1016/j.jfa.2003.07.012. Google Scholar [20] A. Farina and E. Valdinoci, Geometry of quasiminimal phase transitions,, Calc. Var. Part. Diff. Eqs., 33 (2008), 1. doi: 10.1007/s00526-007-0146-1. Google Scholar [21] M. Fazly and N. Ghoussoub, De Giorgi type results for elliptic systems,, Calculus of Variations and Partial Differential Equations, 47 (2013), 809. doi: 10.1007/s00526-012-0536-x. Google Scholar [22] G. Fusco, Equivariant entire solutions to the elliptic system $\Delta u=W_u(u)$ for general $G-$invariant potentials,, Calc. Var. Part. Diff. Eqs., 49 (2014), 963. doi: 10.1007/s00526-013-0607-7. Google Scholar [23] G. Fusco, On some elementary properties of vector minimizers of the Allen-Cahn energy,, Comm. Pure Appl. Anal., 13 (2014), 1045. doi: 10.3934/cpaa.2014.13.1045. Google Scholar [24] G. Fusco, F. Leonetti and C. Pignotti, A uniform estimate for positive solutions of semilinear elliptic equations,, Trans. Amer. Math. Soc., 363 (2011), 4285. doi: 10.1090/S0002-9947-2011-05356-0. Google Scholar [25] E. Gonzalez, U. Massari and I. Tamanini, On the regularity of boundaries of sets minimizing perimeter with a volume constraint,, Indiana Univ. Math. Journal, 32 (1983), 25. doi: 10.1512/iumj.1983.32.32003. Google Scholar [26] C. Gui and M. Schatzman, Symmetric quadruple phase transitions,, Ind. Univ. Math. J., 57 (2008), 781. doi: 10.1512/iumj.2008.57.3089. Google Scholar [27] J. Rubinstein, P. Sternberg and J. Keller, Fast reaction, slow diffusion and curve shortening,, SIAM J. Appl. Math., 49 (1989), 116. doi: 10.1137/0149007. Google Scholar [28] J. Rubinstein, P. Sternberg and J. Keller, Reaction-Diffusion processes and evolution to harmonic maps,, SIAM J. Appl. Math., 49 (1989), 1722. doi: 10.1137/0149104. Google Scholar [29] O. Savin, Regularity of flat level sets in phase transitions,, Ann. of Math., 169 (2009), 41. doi: 10.4007/annals.2009.169.41. Google Scholar [30] O. Savin and E. Valdinoci, Density estimates for a variational model driven by the Gagliardo norm,, Journal de Mathématiques Pures et Appliquées, 101 (2014), 1. doi: 10.1016/j.matpur.2013.05.001. Google Scholar [31] O. Savin and E. Valdinoci, Density estimates for a nonolocal variational model via the Sobolev inequality,, SIAM J. Math. Anal., 43 (2011), 2675. doi: 10.1137/110831040. Google Scholar [32] Y. Sire and E. Valdinoci, Density estimates for phase transitions with a trace,, Interfaces And Free Boundaries, 14 (2012), 153. doi: 10.4171/IFB/277. Google Scholar [33] P. Smyrnelis, Personal, communication., (). Google Scholar [34] P. Sternberg, Vector-valued local minimizers of nonconvex variational problems,, Rocky Mountain J. Math., 21 (1991), 799. doi: 10.1216/rmjm/1181072968. Google Scholar [35] J. E. Taylor, The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces,, Ann. Math., 103 (1976), 489. doi: 10.2307/1970949. Google Scholar [36] E. Valdinoci, Plane-like minimizers in periodic media: Jet flows and Ginzburg-Landau-type functionals,, J. Reine Angew. Math., 574 (2004), 147. doi: 10.1515/crll.2004.068. Google Scholar [37] B. White, Topics in GMT,, Notes by O. Chodash., (2012). Google Scholar
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