American Institute of Mathematical Sciences

Advanced
May  2014, 13(3): 1045-1060. doi: 10.3934/cpaa.2014.13.1045

On some elementary properties of vector minimizers of the Allen-Cahn energy

Giorgio Fusco 1,

1. 

Dipartimento di Matematica Pura e Applicata, Università de L’Aquila, I-67100 L’Aquila

Received  March 2013 Revised  September 2013 Published  December 2013

We derive a point-wise estimate for a map $u: \Omega \subset R^n \rightarrow R^m$ that minimizes $J_A(v): \int_A \frac{1}{2}|\nabla v|^2+U(v)$ subjected to the Dirichlet condition $v=u$ on $\partial\Omega$ for every open smooth and bounded set $A \subset \Omega$. We discuss some consequences of this basic estimate.
Keywords: point-wise estimate., Local minimizer, polar representation, deformation lemma, energy estimates.
Mathematics Subject Classification: Primary: 35J47, 35J50; Secondary: 35J2.
Citation: Giorgio Fusco. On some elementary properties of vector minimizers of the Allen-Cahn energy. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1045-1060. doi: 10.3934/cpaa.2014.13.1045
References:
[1]

G. Alberti, L. Ambrosio and X. Cabré, On a long-standing conjecture of E. De Giorgi: simmetry in 3D for general non linearities and a local minimality property,, \emph{Acta Appl. Math.}, 65 (2001), 9.  doi: 10.1023/A:1010602715526.  Google Scholar

[2]

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

[3]

N. D. Alikakos and G. Fusco, Entire solutions to equivariant elliptic system with variational structure,, \emph{Arch. Rational Mech. Anal.}, 202 (2011), 567.  doi: 10.1007/s00205-011-0441-z.  Google Scholar

[4]

N. D. Alikakos and G. Fusco, Asymptotic rigidity results for symmetric solutions of the elliptic system $\Delta u = Wu(u)$,, work in progress., ().   Google Scholar

[5]

N. D. Alikakos and G. Fusco, A maximum principle for systems with variational structure and an application to standing waves,, preprint, (2012).   Google Scholar

[6]

P. W. Bates, G. Fusco and P. Smyrnelis, Entire solutions with six-fold junctions to elliptic gradient systems with triangle symmetry,, \emph{Advan. Nonlin. Stud.}, 13 (2013), 1.   Google Scholar

[7]

P. W. Bates, G. Fusco and P. Smyrnelis, Multyphase solutions to the vector Allen-Cahn equations: Crystalline and other complex symmetric structures,, work in progress., ().   Google Scholar

[8]

A. Czarnecki, M. Kulczychi and W. Lubawski, On the connectedness of boundary and complement for domains,, \emph{Ann. Polin. Math.}, 103 (2011), 189.  doi: 10.4064/ap103-2-6.  Google Scholar

[9]

G. Fusco, Equivariant entire solutions to the elliptic system $\Delta u=W_u(u)$ for general $G-$invariant potentials,, \emph{Calc. Var. Part. Diff. Eqs.}, (2013), 1.   Google Scholar

[10]

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

[11]

B. Gidas, W. M. Ni and L. Niremberg, Symmetry and related properties via the maximum principle,, \emph{Comm. Math. Phys.}, 68 (1979), 209.  doi: 10.1007/BF01221125.  Google Scholar

[12]

J. Liouville, Lecons sur les fonctions doublement pèriodiques,, \emph{J. Reine Angew. Math.}, 88 (1879), 277.   Google Scholar

show all references

References:
[1]

G. Alberti, L. Ambrosio and X. Cabré, On a long-standing conjecture of E. De Giorgi: simmetry in 3D for general non linearities and a local minimality property,, \emph{Acta Appl. Math.}, 65 (2001), 9.  doi: 10.1023/A:1010602715526.  Google Scholar

[2]

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

[3]

N. D. Alikakos and G. Fusco, Entire solutions to equivariant elliptic system with variational structure,, \emph{Arch. Rational Mech. Anal.}, 202 (2011), 567.  doi: 10.1007/s00205-011-0441-z.  Google Scholar

[4]

N. D. Alikakos and G. Fusco, Asymptotic rigidity results for symmetric solutions of the elliptic system $\Delta u = Wu(u)$,, work in progress., ().   Google Scholar

[5]

N. D. Alikakos and G. Fusco, A maximum principle for systems with variational structure and an application to standing waves,, preprint, (2012).   Google Scholar

[6]

P. W. Bates, G. Fusco and P. Smyrnelis, Entire solutions with six-fold junctions to elliptic gradient systems with triangle symmetry,, \emph{Advan. Nonlin. Stud.}, 13 (2013), 1.   Google Scholar

[7]

P. W. Bates, G. Fusco and P. Smyrnelis, Multyphase solutions to the vector Allen-Cahn equations: Crystalline and other complex symmetric structures,, work in progress., ().   Google Scholar

[8]

A. Czarnecki, M. Kulczychi and W. Lubawski, On the connectedness of boundary and complement for domains,, \emph{Ann. Polin. Math.}, 103 (2011), 189.  doi: 10.4064/ap103-2-6.  Google Scholar

[9]

G. Fusco, Equivariant entire solutions to the elliptic system $\Delta u=W_u(u)$ for general $G-$invariant potentials,, \emph{Calc. Var. Part. Diff. Eqs.}, (2013), 1.   Google Scholar

[10]

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

[11]

B. Gidas, W. M. Ni and L. Niremberg, Symmetry and related properties via the maximum principle,, \emph{Comm. Math. Phys.}, 68 (1979), 209.  doi: 10.1007/BF01221125.  Google Scholar

[12]

J. Liouville, Lecons sur les fonctions doublement pèriodiques,, \emph{J. Reine Angew. Math.}, 88 (1879), 277.   Google Scholar

[1]

Yongqin Liu. The point-wise estimates of solutions for semi-linear dissipative wave equation. Communications on Pure & Applied Analysis, 2013, 12 (1) : 237-252. doi: 10.3934/cpaa.2013.12.237
[2]

Jiao Chen, Weike Wang. The point-wise estimates for the solution of damped wave equation with nonlinear convection in multi-dimensional space. Communications on Pure & Applied Analysis, 2014, 13 (1) : 307-330. doi: 10.3934/cpaa.2014.13.307
[3]

Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial & Management Optimization, 2017, 13 (5) : 0-0. doi: 10.3934/jimo.2019115
[4]

Petteri Harjulehto, Peter Hästö, Juha Tiirola. Point-wise behavior of the Geman--McClure and the Hebert--Leahy image restoration models. Inverse Problems & Imaging, 2015, 9 (3) : 835-851. doi: 10.3934/ipi.2015.9.835
[5]

Erik Ekström, Johan Tysk. A boundary point lemma for Black-Scholes type operators. Communications on Pure & Applied Analysis, 2006, 5 (3) : 505-514. doi: 10.3934/cpaa.2006.5.505
[6]

Tohru Nakamura, Shinya Nishibata. Energy estimate for a linear symmetric hyperbolic-parabolic system in half line. Kinetic & Related Models, 2013, 6 (4) : 883-892. doi: 10.3934/krm.2013.6.883
[7]

Boris P. Belinskiy, Peter Caithamer. Energy estimate for the wave equation driven by a fractional Gaussian noise. Conference Publications, 2007, 2007 (Special) : 92-101. doi: 10.3934/proc.2007.2007.92
[8]

Rui Qian, Rong Hu, Ya-Ping Fang. Local smooth representation of solution sets in parametric linear fractional programming problems. Numerical Algebra, Control & Optimization, 2019, 9 (1) : 45-52. doi: 10.3934/naco.2019004
[9]

Xavier Perrot, Xavier Carton. Point-vortex interaction in an oscillatory deformation field: Hamiltonian dynamics, harmonic resonance and transition to chaos. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 971-995. doi: 10.3934/dcdsb.2009.11.971
[10]

Rongjie Lai, Jiang Liang, Hong-Kai Zhao. A local mesh method for solving PDEs on point clouds. Inverse Problems & Imaging, 2013, 7 (3) : 737-755. doi: 10.3934/ipi.2013.7.737
[11]

Noam Presman, Simon Litsyn. Recursive descriptions of polar codes. Advances in Mathematics of Communications, 2017, 11 (1) : 1-65. doi: 10.3934/amc.2017001
[12]

L.R. Ritter, Akif Ibragimov, Jay R. Walton, Catherine J. McNeal. Stability analysis using an energy estimate approach of a reaction-diffusion model of atherogenesis. Conference Publications, 2009, 2009 (Special) : 630-639. doi: 10.3934/proc.2009.2009.630
[13]

J. F. Padial. Existence and estimate of the location of the free-boundary for a non local inverse elliptic-parabolic problem arising in nuclear fusion. Conference Publications, 2011, 2011 (Special) : 1176-1185. doi: 10.3934/proc.2011.2011.1176
[14]

Riccardo Adami, Diego Noja, Nicola Visciglia. Constrained energy minimization and ground states for NLS with point defects. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1155-1188. doi: 10.3934/dcdsb.2013.18.1155
[15]

Gautier Picot. Energy-minimal transfers in the vicinity of the lagrangian point $L_1$. Conference Publications, 2011, 2011 (Special) : 1196-1205. doi: 10.3934/proc.2011.2011.1196
[16]

Jaeyoung Byeon, Sungwon Cho, Junsang Park. On the location of a peak point of a least energy solution for Hénon equation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1055-1081. doi: 10.3934/dcds.2011.30.1055
[17]

Boris Andreianov, Carlotta Donadello, Ulrich Razafison, Massimiliano D. Rosini. Riemann problems with non--local point constraints and capacity drop. Mathematical Biosciences & Engineering, 2015, 12 (2) : 259-278. doi: 10.3934/mbe.2015.12.259
[18]

Mohamed Benyahia, Massimiliano D. Rosini. A macroscopic traffic model with phase transitions and local point constraints on the flow. Networks & Heterogeneous Media, 2017, 12 (2) : 297-317. doi: 10.3934/nhm.2017013
[19]

Annegret Glitzky. Energy estimates for electro-reaction-diffusion systems with partly fast kinetics. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 159-174. doi: 10.3934/dcds.2009.25.159
[20]

Dorina Mitrea, Irina Mitrea, Marius Mitrea, Lixin Yan. Coercive energy estimates for differential forms in semi-convex domains. Communications on Pure & Applied Analysis, 2010, 9 (4) : 987-1010. doi: 10.3934/cpaa.2010.9.987

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (25)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences