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

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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

Abstract
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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.

Keywords: monotonicity formula, Allen-Cahn equation, Entire solutions, phase transitions.

Mathematics Subject Classification: Primary: 35J20, 35J91; Secondary: 35J4.

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, ().

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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, ().

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