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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

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. Google Scholar [2] N. D. Alikakos and G. Fusco, Density estimates for vector minimizers and applications,, preprint, (). Google Scholar [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. Google Scholar [4] P. W. Bates, G. Fusco and P. Smyrnelis, Multiphase solutions to the vector Allen-Cahn equation: crystalline and other complex symmetric structures,, preprint, (). Google Scholar [5] F. Bethuel, H. Brezis and G. Orlandi, Asymptotics of the Ginzburg-Landau equation in arbitrary dimensions,, \emph{J. Funct. Anal.}, 186 (2001), 432. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [15] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, 2$^{nd}$ edition, (1983). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [20] T. Riviére, Asymptotic analysis for the Ginzburg-Landau equations,, \emph{Bollettino dell'Unione Matematica Italiana 2-B}, 3 (1999), 537. Google Scholar [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. Google Scholar [22] P. Smyrnelis, Gradient estimates for semilinear elliptic systems and other related results,, preprint, (). Google Scholar [23] R. Sperb, Maximum Principles and their Applications,, Academic Press, (1981). Google Scholar [24] C. Sourdis, Optimal energy growth lower bounds for a class of solutions to the vectorial Allen-Cahn equation,, preprint, (). Google Scholar [25] C. Sourdis, A new monotonicity formula for solutions to the elliptic system $\Delta u=\nabla W(u)$,, preprint, (). Google Scholar [26] C. Sourdis, Energy estimates for minimizers to a class of elliptic systems of Allen-Cahn type and the Liouville property,, preprint, (). Google Scholar

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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. Google Scholar [2] N. D. Alikakos and G. Fusco, Density estimates for vector minimizers and applications,, preprint, (). Google Scholar [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. Google Scholar [4] P. W. Bates, G. Fusco and P. Smyrnelis, Multiphase solutions to the vector Allen-Cahn equation: crystalline and other complex symmetric structures,, preprint, (). Google Scholar [5] F. Bethuel, H. Brezis and G. Orlandi, Asymptotics of the Ginzburg-Landau equation in arbitrary dimensions,, \emph{J. Funct. Anal.}, 186 (2001), 432. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [15] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, 2$^{nd}$ edition, (1983). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [20] T. Riviére, Asymptotic analysis for the Ginzburg-Landau equations,, \emph{Bollettino dell'Unione Matematica Italiana 2-B}, 3 (1999), 537. Google Scholar [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. Google Scholar [22] P. Smyrnelis, Gradient estimates for semilinear elliptic systems and other related results,, preprint, (). Google Scholar [23] R. Sperb, Maximum Principles and their Applications,, Academic Press, (1981). Google Scholar [24] C. Sourdis, Optimal energy growth lower bounds for a class of solutions to the vectorial Allen-Cahn equation,, preprint, (). Google Scholar [25] C. Sourdis, A new monotonicity formula for solutions to the elliptic system $\Delta u=\nabla W(u)$,, preprint, (). Google Scholar [26] C. Sourdis, Energy estimates for minimizers to a class of elliptic systems of Allen-Cahn type and the Liouville property,, preprint, (). Google Scholar
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