• Previous Article
    The Liouville type theorem and local regularity results for nonlinear differential and integral systems
  • CPAA Home
  • This Issue
  • Next Article
    Note on global regularity of 3D generalized magnetohydrodynamic-$\alpha$ model with zero diffusivity
March  2015, 14(2): 577-584. doi: 10.3934/cpaa.2015.14.577

On the growth of the energy of entire solutions to the vector Allen-Cahn equation

1. 

Department of Mathematics and Applied Mathematics, University of Crete, Voutes University Campus, Heraklion 71003, China

Received  April 2014 Revised  September 2014 Published  December 2014

We prove that the energy over balls of entire, nonconstant bounded solutions to the vector Allen-Cahn equation grows faster than $(\ln R)^k R^{n-2}$, for any $k>0$, as the radius $R$ of the $n$-dimensional ball tends to infinity. This improves the growth rate of order $R^{n-2}$ if $n\geq 3$ and $\ln R$ if $n=2$ that follows from the general weak monotonicity formula. Moreover, our estimate may be considered as an approximation to the corresponding rate of order $R^{n-1}$ that is known to hold in the scalar case.
Citation: Christos Sourdis. On the growth of the energy of entire solutions to the vector Allen-Cahn equation. Communications on Pure & Applied Analysis, 2015, 14 (2) : 577-584. doi: 10.3934/cpaa.2015.14.577
References:
[1]

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.

[2]

N. D. Alikakos and G. Fusco, Density estimates for vector minimizers and applications,, preprint, ().

[3]

S. Baldo, Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 7 (1990), 67.

[4]

P. W. Bates, G. Fusco and P. Smyrnelis, Multiphase solutions to the vector Allen-Cahn equation: crystalline and other complex symmetric structures,, preprint, ().

[5]

F. Bethuel, H. Brezis and G. Orlandi, Asymptotics of the Ginzburg-Landau equation in arbitrary dimensions,, \emph{J. Funct. Anal.}, 186 (2001), 432.

[6]

L. Bronsard and F. Reitich, On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation,, \emph{Arch. Ration. Mech. Anal.}, 124 (1993), 355. doi: 10.1007/BF00375607.

[7]

L. Caffarelli, N. Garofalo and F. Segála, A gradient bound for entire solutions of quasi-linear equations and its consequences,, \emph{Comm. Pure Appl. Math.}, 47 (1994), 1457.

[8]

L. Caffarelli and A. Córdoba, Uniform convergence of a singular perturbation problem,, \emph{Comm. Pure Appl. Math.}, 48 (1995), 1. doi: 10.1002/cpa.3160480101.

[9]

L. A. Caffarelli and F-H. Lin, Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries,, \emph{J. Amer. Math. Soc.}, 21 (2008), 847. doi: 10.1090/S0894-0347-08-00593-6.

[10]

M. del Pino, P. Felmer and M. Kowalczyk, Minimality and nondegeneracy of degree-one Ginzburg-Landau vortex as a Hardy's type inequality,, \emph{Int. Math. Res. Not.}, 30 (2004), 1511. doi: 10.1155/S1073792804133588.

[11]

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

[12]

A. Farina and E. Valdinoci, Flattening results for elliptic PDEs in unbounded domains with applications to overdetermined problems,, \emph{Arch. Ration. Mech. Anal.}, 195 (2010), 1025. doi: 10.1007/s00205-009-0227-8.

[13]

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

[14]

N. Ghoussoub and B. Pass, Decoupling of De Giorgi-type systems via multi-marginal optimal transport,, \emph{Comm. Partial Differential Equations}, 39 (2014), 1032. doi: 10.1080/03605302.2013.849730.

[15]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, 2$^{nd}$ edition, (1983).

[16]

C. Gui and F. Zhou, Asymptotic behavior of oscillating radial solutions to certain nonlinear equations,, \emph{Methods Appl. Anal.}, 15 (2008), 285. doi: 10.4310/MAA.2008.v15.n3.a3.

[17]

P. Mironescu and V. Radulescu, Periodic solutions of the equation $-\Delta v=v(1-|v|^2)$ in $\mathbbR$ and $\mathbbR^2$,, \emph{Houston J. Math.}, 20 (1994), 653.

[18]

L. Modica, A gradient bound and a Liouville theorem for nonlinear Poisson equations,, \emph{Comm. Pure Appl. Math.}, 38 (1985), 679. doi: 10.1002/cpa.3160380515.

[19]

L. Modica, Monotonicity of the energy for entire solutions of semilinear elliptic equations,, in \emph{Partial Differential Equations and the Calculus of Variations, (1989), 843.

[20]

T. Riviére, Asymptotic analysis for the Ginzburg-Landau equations,, \emph{Bollettino dell'Unione Matematica Italiana 2-B}, 3 (1999), 537.

[21]

D. Smets, PDE analysis of concentrating energies for the Ginzburg-Landau equation,, in \emph{Topics on Concentration Phenomena and Problems with Multiple Scales} (eds. A. Braides and V. C. Piat), (2006), 293. doi: 10.1007/978-3-540-36546-4_8.

[22]

P. Smyrnelis, Gradient estimates for semilinear elliptic systems and other related results,, preprint, ().

[23]

R. Sperb, Maximum Principles and their Applications,, Academic Press, (1981).

[24]

C. Sourdis, Optimal energy growth lower bounds for a class of solutions to the vectorial Allen-Cahn equation,, preprint, ().

[25]

C. Sourdis, A new monotonicity formula for solutions to the elliptic system $\Delta u=\nabla W(u)$,, preprint, ().

[26]

C. Sourdis, Energy estimates for minimizers to a class of elliptic systems of Allen-Cahn type and the Liouville property,, preprint, ().

show all references

References:
[1]

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.

[2]

N. D. Alikakos and G. Fusco, Density estimates for vector minimizers and applications,, preprint, ().

[3]

S. Baldo, Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 7 (1990), 67.

[4]

P. W. Bates, G. Fusco and P. Smyrnelis, Multiphase solutions to the vector Allen-Cahn equation: crystalline and other complex symmetric structures,, preprint, ().

[5]

F. Bethuel, H. Brezis and G. Orlandi, Asymptotics of the Ginzburg-Landau equation in arbitrary dimensions,, \emph{J. Funct. Anal.}, 186 (2001), 432.

[6]

L. Bronsard and F. Reitich, On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation,, \emph{Arch. Ration. Mech. Anal.}, 124 (1993), 355. doi: 10.1007/BF00375607.

[7]

L. Caffarelli, N. Garofalo and F. Segála, A gradient bound for entire solutions of quasi-linear equations and its consequences,, \emph{Comm. Pure Appl. Math.}, 47 (1994), 1457.

[8]

L. Caffarelli and A. Córdoba, Uniform convergence of a singular perturbation problem,, \emph{Comm. Pure Appl. Math.}, 48 (1995), 1. doi: 10.1002/cpa.3160480101.

[9]

L. A. Caffarelli and F-H. Lin, Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries,, \emph{J. Amer. Math. Soc.}, 21 (2008), 847. doi: 10.1090/S0894-0347-08-00593-6.

[10]

M. del Pino, P. Felmer and M. Kowalczyk, Minimality and nondegeneracy of degree-one Ginzburg-Landau vortex as a Hardy's type inequality,, \emph{Int. Math. Res. Not.}, 30 (2004), 1511. doi: 10.1155/S1073792804133588.

[11]

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

[12]

A. Farina and E. Valdinoci, Flattening results for elliptic PDEs in unbounded domains with applications to overdetermined problems,, \emph{Arch. Ration. Mech. Anal.}, 195 (2010), 1025. doi: 10.1007/s00205-009-0227-8.

[13]

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

[14]

N. Ghoussoub and B. Pass, Decoupling of De Giorgi-type systems via multi-marginal optimal transport,, \emph{Comm. Partial Differential Equations}, 39 (2014), 1032. doi: 10.1080/03605302.2013.849730.

[15]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, 2$^{nd}$ edition, (1983).

[16]

C. Gui and F. Zhou, Asymptotic behavior of oscillating radial solutions to certain nonlinear equations,, \emph{Methods Appl. Anal.}, 15 (2008), 285. doi: 10.4310/MAA.2008.v15.n3.a3.

[17]

P. Mironescu and V. Radulescu, Periodic solutions of the equation $-\Delta v=v(1-|v|^2)$ in $\mathbbR$ and $\mathbbR^2$,, \emph{Houston J. Math.}, 20 (1994), 653.

[18]

L. Modica, A gradient bound and a Liouville theorem for nonlinear Poisson equations,, \emph{Comm. Pure Appl. Math.}, 38 (1985), 679. doi: 10.1002/cpa.3160380515.

[19]

L. Modica, Monotonicity of the energy for entire solutions of semilinear elliptic equations,, in \emph{Partial Differential Equations and the Calculus of Variations, (1989), 843.

[20]

T. Riviére, Asymptotic analysis for the Ginzburg-Landau equations,, \emph{Bollettino dell'Unione Matematica Italiana 2-B}, 3 (1999), 537.

[21]

D. Smets, PDE analysis of concentrating energies for the Ginzburg-Landau equation,, in \emph{Topics on Concentration Phenomena and Problems with Multiple Scales} (eds. A. Braides and V. C. Piat), (2006), 293. doi: 10.1007/978-3-540-36546-4_8.

[22]

P. Smyrnelis, Gradient estimates for semilinear elliptic systems and other related results,, preprint, ().

[23]

R. Sperb, Maximum Principles and their Applications,, Academic Press, (1981).

[24]

C. Sourdis, Optimal energy growth lower bounds for a class of solutions to the vectorial Allen-Cahn equation,, preprint, ().

[25]

C. Sourdis, A new monotonicity formula for solutions to the elliptic system $\Delta u=\nabla W(u)$,, preprint, ().

[26]

C. Sourdis, Energy estimates for minimizers to a class of elliptic systems of Allen-Cahn type and the Liouville property,, preprint, ().

[1]

Hirokazu Ninomiya. Entire solutions and traveling wave solutions of the Allen-Cahn-Nagumo equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2001-2019. doi: 10.3934/dcds.2019084

[2]

Isabeau Birindelli, Enrico Valdinoci. On the Allen-Cahn equation in the Grushin plane: A monotone entire solution that is not one-dimensional. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 823-838. doi: 10.3934/dcds.2011.29.823

[3]

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

[4]

Eleonora Cinti. Saddle-shaped solutions for the fractional Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 441-463. doi: 10.3934/dcdss.2018024

[5]

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

[6]

Jun Yang, Xiaolin Yang. Clustered interior phase transition layers for an inhomogeneous Allen-Cahn equation in higher dimensional domains. Communications on Pure & Applied Analysis, 2013, 12 (1) : 303-340. doi: 10.3934/cpaa.2013.12.303

[7]

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

[8]

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

[9]

Giorgio Fusco, Francesco Leonetti, Cristina Pignotti. On the asymptotic behavior of symmetric solutions of the Allen-Cahn equation in unbounded domains in $\mathbb{R}^2$. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 725-742. doi: 10.3934/dcds.2017030

[10]

Michał Kowalczyk, Yong Liu, Frank Pacard. Towards classification of multiple-end solutions to the Allen-Cahn equation in $\mathbb{R}^2$. Networks & Heterogeneous Media, 2012, 7 (4) : 837-855. doi: 10.3934/nhm.2012.7.837

[11]

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

[12]

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

[13]

Ciprian G. Gal, Maurizio Grasselli. The non-isothermal Allen-Cahn equation with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 1009-1040. doi: 10.3934/dcds.2008.22.1009

[14]

Zhuoran Du, Baishun Lai. Transition layers for an inhomogeneous Allen-Cahn equation in Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1407-1429. doi: 10.3934/dcds.2013.33.1407

[15]

Charles-Edouard Bréhier, Ludovic Goudenège. Analysis of some splitting schemes for the stochastic Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-22. doi: 10.3934/dcdsb.2019077

[16]

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

[17]

Changchun Liu, Hui Tang. Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. Evolution Equations & Control Theory, 2017, 6 (2) : 219-237. doi: 10.3934/eect.2017012

[18]

Cristina Pocci. On singular limit of a nonlinear $p$-order equation related to Cahn-Hilliard and Allen-Cahn evolutions. Evolution Equations & Control Theory, 2013, 2 (3) : 517-530. doi: 10.3934/eect.2013.2.517

[19]

Ahmad Makki, Alain Miranville. Existence of solutions for anisotropic Cahn-Hilliard and Allen-Cahn systems in higher space dimensions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 759-775. doi: 10.3934/dcdss.2016027

[20]

Fang Li, Kimie Nakashima. Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1391-1420. doi: 10.3934/dcds.2012.32.1391

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]