American Institute of Mathematical Sciences

Advanced
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 and Continuous Dynamical Systems, 2015, 35 (12) : 5631-5663. doi: 10.3934/dcds.2015.35.5631
References:
[1]

S. Alama, L. Bronsard and C. Gui, Stationary solutions in $\mathbb{R}^2$ for an Allen-Cahn system with multiple well potential, Calc. Var. Part. Diff. Eqs., 5 (1997), 359-390. doi: 10.1007/s005260050071.

[2]

N. D. Alikakos, Some basic facts on the system $\Delta u-W_u(u)=0$, Proc. Amer. Math. Soc., 139 (2011), 153-162. doi: 10.1090/S0002-9939-2010-10453-7.

[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), Publications Scuola Normale Superiore, CRM Series, 15, Birkhäuser, 2013, 1-31. doi: 10.1007/978-88-7642-473-1_1.

[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), 2093-2115. doi: 10.1080/03605302.2012.721851.

[5]

N. D. Alikakos and G. Fusco, Entire solutions to equivariant elliptic systems with variational structure, Arch. Rat. Mech. Analysis, 202 (2011), 567-597. doi: 10.1007/s00205-011-0441-z.

[6]

N. D. Alikakos and G. Fusco, In preparation.

[7]

N. D. Alikakos and G. Fusco, On the connection problem for potentials with several global minima, Indiana Univ. Math. Journal, 57 (2008), 1871-1906. doi: 10.1512/iumj.2008.57.3181.

[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), 1-26.

[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, arXiv:1311.1022.

[10]

S. Baldo, Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids, Ann. Inst. Henri Poincare, 7 (1990), 67-90.

[11]

F. Bethuel, H. Brezis and F. Helein, Ginzburg-Landau Vortices, Birkhäuser, 1994. doi: 10.1007/978-1-4612-0287-5.

[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), 355-379. doi: 10.1007/BF00375607.

[13]

L. Bronsard, C. Gui and M. Schatzman, A three-layered minimizer in $\mathbb{R}^2$ for a variational problem with a symmetric three-well potential, 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.

[14]

L. Caffarelli and A. Cordoba, Uniform convergence of a singular perturbation problem, Comm. Pure Appl. Math., 48 (1995), 1-12. doi: 10.1002/cpa.3160480101.

[15]

L. Caffarelli and F. Lin, Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries, Journal Amer. Math. Society , 21 (2008), 847-862. doi: 10.1090/S0894-0347-08-00593-6.

[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), 153-191. doi: 10.1007/s00205-014-0804-3.

[17]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, AMS, 1998. doi: 10.1090/gsm/019.

[18]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.

[19]

A. Farina, Two results on entire solutions of Ginzburg-Landau systems in higher dimensions, J. Funct. Anal., 214 (2004), 386-395. doi: 10.1016/j.jfa.2003.07.012.

[20]

A. Farina and E. Valdinoci, Geometry of quasiminimal phase transitions, Calc. Var. Part. Diff. Eqs., 33 (2008), 1-35. doi: 10.1007/s00526-007-0146-1.

[21]

M. Fazly and N. Ghoussoub, De Giorgi type results for elliptic systems, Calculus of Variations and Partial Differential Equations, 47 (2013), 809-823. doi: 10.1007/s00526-012-0536-x.

[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), 963-985. doi: 10.1007/s00526-013-0607-7.

[23]

G. Fusco, On some elementary properties of vector minimizers of the Allen-Cahn energy, Comm. Pure Appl. Anal., 13 (2014), 1045-1060. doi: 10.3934/cpaa.2014.13.1045.

[24]

G. Fusco, F. Leonetti and C. Pignotti, A uniform estimate for positive solutions of semilinear elliptic equations, Trans. Amer. Math. Soc., 363 (2011), 4285-4307. doi: 10.1090/S0002-9947-2011-05356-0.

[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), 25-37. doi: 10.1512/iumj.1983.32.32003.

[26]

C. Gui and M. Schatzman, Symmetric quadruple phase transitions, Ind. Univ. Math. J., 57 (2008), 781-836. doi: 10.1512/iumj.2008.57.3089.

[27]

J. Rubinstein, P. Sternberg and J. Keller, Fast reaction, slow diffusion and curve shortening, SIAM J. Appl. Math., 49 (1989), 116-133. doi: 10.1137/0149007.

[28]

J. Rubinstein, P. Sternberg and J. Keller, Reaction-Diffusion processes and evolution to harmonic maps, SIAM J. Appl. Math., 49 (1989), 1722-1733. doi: 10.1137/0149104.

[29]

O. Savin, Regularity of flat level sets in phase transitions, Ann. of Math., 169 (2009), 41-78. doi: 10.4007/annals.2009.169.41.

[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), 1-26. doi: 10.1016/j.matpur.2013.05.001.

[31]

O. Savin and E. Valdinoci, Density estimates for a nonolocal variational model via the Sobolev inequality, SIAM J. Math. Anal., 43 (2011), 2675-2687. doi: 10.1137/110831040.

[32]

Y. Sire and E. Valdinoci, Density estimates for phase transitions with a trace, Interfaces And Free Boundaries, 14 (2012), 153-165. doi: 10.4171/IFB/277.

[33]

P. Smyrnelis, Personal communication.

[34]

P. Sternberg, Vector-valued local minimizers of nonconvex variational problems, Rocky Mountain J. Math., 21 (1991), 799-807. doi: 10.1216/rmjm/1181072968.

[35]

J. E. Taylor, The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces, Ann. Math., 103 (1976), 489-539. doi: 10.2307/1970949.

[36]

E. Valdinoci, Plane-like minimizers in periodic media: Jet flows and Ginzburg-Landau-type functionals, J. Reine Angew. Math., 574 (2004), 147-185. doi: 10.1515/crll.2004.068.

[37]

B. White, Topics in GMT, Notes by O. Chodash., Stanford, 2012.

show all references

References:
[1]

S. Alama, L. Bronsard and C. Gui, Stationary solutions in $\mathbb{R}^2$ for an Allen-Cahn system with multiple well potential, Calc. Var. Part. Diff. Eqs., 5 (1997), 359-390. doi: 10.1007/s005260050071.

[2]

N. D. Alikakos, Some basic facts on the system $\Delta u-W_u(u)=0$, Proc. Amer. Math. Soc., 139 (2011), 153-162. doi: 10.1090/S0002-9939-2010-10453-7.

[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), Publications Scuola Normale Superiore, CRM Series, 15, Birkhäuser, 2013, 1-31. doi: 10.1007/978-88-7642-473-1_1.

[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), 2093-2115. doi: 10.1080/03605302.2012.721851.

[5]

N. D. Alikakos and G. Fusco, Entire solutions to equivariant elliptic systems with variational structure, Arch. Rat. Mech. Analysis, 202 (2011), 567-597. doi: 10.1007/s00205-011-0441-z.

[6]

N. D. Alikakos and G. Fusco, In preparation.

[7]

N. D. Alikakos and G. Fusco, On the connection problem for potentials with several global minima, Indiana Univ. Math. Journal, 57 (2008), 1871-1906. doi: 10.1512/iumj.2008.57.3181.

[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), 1-26.

[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, arXiv:1311.1022.

[10]

S. Baldo, Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids, Ann. Inst. Henri Poincare, 7 (1990), 67-90.

[11]

F. Bethuel, H. Brezis and F. Helein, Ginzburg-Landau Vortices, Birkhäuser, 1994. doi: 10.1007/978-1-4612-0287-5.

[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), 355-379. doi: 10.1007/BF00375607.

[13]

L. Bronsard, C. Gui and M. Schatzman, A three-layered minimizer in $\mathbb{R}^2$ for a variational problem with a symmetric three-well potential, 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.

[14]

L. Caffarelli and A. Cordoba, Uniform convergence of a singular perturbation problem, Comm. Pure Appl. Math., 48 (1995), 1-12. doi: 10.1002/cpa.3160480101.

[15]

L. Caffarelli and F. Lin, Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries, Journal Amer. Math. Society , 21 (2008), 847-862. doi: 10.1090/S0894-0347-08-00593-6.

[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), 153-191. doi: 10.1007/s00205-014-0804-3.

[17]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, AMS, 1998. doi: 10.1090/gsm/019.

[18]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.

[19]

A. Farina, Two results on entire solutions of Ginzburg-Landau systems in higher dimensions, J. Funct. Anal., 214 (2004), 386-395. doi: 10.1016/j.jfa.2003.07.012.

[20]

A. Farina and E. Valdinoci, Geometry of quasiminimal phase transitions, Calc. Var. Part. Diff. Eqs., 33 (2008), 1-35. doi: 10.1007/s00526-007-0146-1.

[21]

M. Fazly and N. Ghoussoub, De Giorgi type results for elliptic systems, Calculus of Variations and Partial Differential Equations, 47 (2013), 809-823. doi: 10.1007/s00526-012-0536-x.

[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), 963-985. doi: 10.1007/s00526-013-0607-7.

[23]

G. Fusco, On some elementary properties of vector minimizers of the Allen-Cahn energy, Comm. Pure Appl. Anal., 13 (2014), 1045-1060. doi: 10.3934/cpaa.2014.13.1045.

[24]

G. Fusco, F. Leonetti and C. Pignotti, A uniform estimate for positive solutions of semilinear elliptic equations, Trans. Amer. Math. Soc., 363 (2011), 4285-4307. doi: 10.1090/S0002-9947-2011-05356-0.

[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), 25-37. doi: 10.1512/iumj.1983.32.32003.

[26]

C. Gui and M. Schatzman, Symmetric quadruple phase transitions, Ind. Univ. Math. J., 57 (2008), 781-836. doi: 10.1512/iumj.2008.57.3089.

[27]

J. Rubinstein, P. Sternberg and J. Keller, Fast reaction, slow diffusion and curve shortening, SIAM J. Appl. Math., 49 (1989), 116-133. doi: 10.1137/0149007.

[28]

J. Rubinstein, P. Sternberg and J. Keller, Reaction-Diffusion processes and evolution to harmonic maps, SIAM J. Appl. Math., 49 (1989), 1722-1733. doi: 10.1137/0149104.

[29]

O. Savin, Regularity of flat level sets in phase transitions, Ann. of Math., 169 (2009), 41-78. doi: 10.4007/annals.2009.169.41.

[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), 1-26. doi: 10.1016/j.matpur.2013.05.001.

[31]

O. Savin and E. Valdinoci, Density estimates for a nonolocal variational model via the Sobolev inequality, SIAM J. Math. Anal., 43 (2011), 2675-2687. doi: 10.1137/110831040.

[32]

Y. Sire and E. Valdinoci, Density estimates for phase transitions with a trace, Interfaces And Free Boundaries, 14 (2012), 153-165. doi: 10.4171/IFB/277.

[33]

P. Smyrnelis, Personal communication.

[34]

P. Sternberg, Vector-valued local minimizers of nonconvex variational problems, Rocky Mountain J. Math., 21 (1991), 799-807. doi: 10.1216/rmjm/1181072968.

[35]

J. E. Taylor, The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces, Ann. Math., 103 (1976), 489-539. doi: 10.2307/1970949.

[36]

E. Valdinoci, Plane-like minimizers in periodic media: Jet flows and Ginzburg-Landau-type functionals, J. Reine Angew. Math., 574 (2004), 147-185. doi: 10.1515/crll.2004.068.

[37]

B. White, Topics in GMT, Notes by O. Chodash., Stanford, 2012.

[1]

Giorgio Fusco. On some elementary properties of vector minimizers of the Allen-Cahn energy. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1045-1060. doi: 10.3934/cpaa.2014.13.1045
[2]

Giorgio Fusco. Layered solutions to the vector Allen-Cahn equation in $\mathbb{R}^2$. Minimizers and heteroclinic connections. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1807-1841. doi: 10.3934/cpaa.2017088
[3]

Christos Sourdis. On the growth of the energy of entire solutions to the vector Allen-Cahn equation. Communications on Pure and Applied Analysis, 2015, 14 (2) : 577-584. doi: 10.3934/cpaa.2015.14.577
[4]

Gianni Gilardi. On an Allen-Cahn type integrodifferential equation. Discrete and Continuous Dynamical Systems - S, 2013, 6 (3) : 703-709. doi: 10.3934/dcdss.2013.6.703
[5]

Quan Wang, Dongming Yan. On the stability and transition of the Cahn-Hilliard/Allen-Cahn system. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2607-2620. doi: 10.3934/dcdsb.2020024
[6]

Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete and Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127
[7]

Christopher P. Grant. Grain sizes in the discrete Allen-Cahn and Cahn-Hilliard equations. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 127-146. doi: 10.3934/dcds.2001.7.127
[8]

Alain Miranville, Ramon Quintanilla, Wafa Saoud. Asymptotic behavior of a Cahn-Hilliard/Allen-Cahn system with temperature. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2257-2288. doi: 10.3934/cpaa.2020099
[9]

Shixing Li, Dongming Yan. On the steady state bifurcation of the Cahn-Hilliard/Allen-Cahn system. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3077-3088. doi: 10.3934/dcdsb.2018301
[10]

Jie Shen, Xiaofeng Yang. Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1669-1691. doi: 10.3934/dcds.2010.28.1669
[11]

Alain Miranville, Wafa Saoud, Raafat Talhouk. On the Cahn-Hilliard/Allen-Cahn equations with singular potentials. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3633-3651. doi: 10.3934/dcdsb.2018308
[12]

Annalisa Cesaroni, Serena Dipierro, Matteo Novaga, Enrico Valdinoci. Minimizers of the $ p $-oscillation functional. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 6785-6799. doi: 10.3934/dcds.2019231
[13]

Florian Krügel. Some properties of minimizers of a variational problem involving the total variation functional. Communications on Pure and Applied Analysis, 2015, 14 (1) : 341-360. doi: 10.3934/cpaa.2015.14.341
[14]

Riccardo March, Giuseppe Riey. Euler equations and trace properties of minimizers of a functional for motion compensated inpainting. Inverse Problems and Imaging, 2022, 16 (4) : 703-737. doi: 10.3934/ipi.2021072
[15]

Hongmei Cheng, Rong Yuan. Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation. Discrete and Continuous Dynamical Systems - B, 2015, 20 (4) : 1015-1029. doi: 10.3934/dcdsb.2015.20.1015
[16]

Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control and Optimization, 2021, 11 (2) : 269-281. doi: 10.3934/naco.2020025
[17]

Tatsuki Mori, Kousuke Kuto, Tohru Tsujikawa, Shoji Yotsutani. Representation formulas of solutions and bifurcation sheets to a nonlocal Allen-Cahn equation. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4907-4925. doi: 10.3934/dcds.2020205
[18]

Xinlong Feng, Huailing Song, Tao Tang, Jiang Yang. Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation. Inverse Problems and Imaging, 2013, 7 (3) : 679-695. doi: 10.3934/ipi.2013.7.679
[19]

Yan Hu. Layer solutions for an Allen-Cahn type system driven by the fractional Laplacian. Communications on Pure and Applied Analysis, 2016, 15 (3) : 947-964. doi: 10.3934/cpaa.2016.15.947
[20]

Paul H. Rabinowitz, Ed Stredulinsky. On a class of infinite transition solutions for an Allen-Cahn model equation. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 319-332. doi: 10.3934/dcds.2008.21.319

2021 Impact Factor: 1.588

Tools

Metrics

  • PDF downloads (51)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy