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The Liouville type theorem and local regularity results for nonlinear differential and integral systems
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 |
References:
[1] |
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. |
[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, Ann. Inst. H. Poincaré Anal. Non Linéaire, 7 (1990), 67-90. |
[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, J. Funct. Anal., 186 (2001), 432-520. |
[6] |
L. Bronsard and F. Reitich, On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation, Arch. Ration. Mech. Anal., 124 (1993), 355-379.
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, Comm. Pure Appl. Math., 47 (1994), 1457-1473. |
[8] |
L. Caffarelli and A. Córdoba, Uniform convergence of a singular perturbation problem, Comm. Pure Appl. Math., 48 (1995), 1-12.
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, J. Amer. Math. Soc., 21 (2008), 847-862.
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, Int. Math. Res. Not., 30 (2004), 1511-1527.
doi: 10.1155/S1073792804133588. |
[11] |
A. Farina, Two results on entire solutions of Ginzburg-Landau system in higher dimensions, J. Funct. Anal., 214 (2004), 386-395.
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, Arch. Ration. Mech. Anal., 195 (2010), 1025-1058.
doi: 10.1007/s00205-009-0227-8. |
[13] |
G. Fusco, On some elementary properties of vector minimizers of the Allen-Cahn energy, Comm. Pure Appl. Anal., 13 (2014), 1045-1060.
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, Comm. Partial Differential Equations, 39 (2014), 1032-1047.
doi: 10.1080/03605302.2013.849730. |
[15] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer-Verlag, New York, 1983. |
[16] |
C. Gui and F. Zhou, Asymptotic behavior of oscillating radial solutions to certain nonlinear equations, Methods Appl. Anal., 15 (2008), 285-296.
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 $\mathbbR^2$, Houston J. Math., 20 (1994), 653-669. |
[18] |
L. Modica, A gradient bound and a Liouville theorem for nonlinear Poisson equations, Comm. Pure Appl. Math., 38 (1985), 679-684.
doi: 10.1002/cpa.3160380515. |
[19] |
L. Modica, Monotonicity of the energy for entire solutions of semilinear elliptic equations, in Partial Differential Equations and the Calculus of Variations, Essays in honor of Ennio De Giorgi, Vol. 2 (eds. F. Colombini, A. Marino and L. Modica), Birkhäuser, (1989), 843-850. |
[20] |
T. Riviére, Asymptotic analysis for the Ginzburg-Landau equations, Bollettino dell'Unione Matematica Italiana 2-B, 3 (1999), 537-575. |
[21] |
D. Smets, PDE analysis of concentrating energies for the Ginzburg-Landau equation, in Topics on Concentration Phenomena and Problems with Multiple Scales (eds. A. Braides and V. C. Piat), Springer, (2006), 293-314.
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, New York, 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 $\Delta u - W_u(u) = 0$, Proc. Amer. Math. Soc., 139 (2011), 153-162.
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, Ann. Inst. H. Poincaré Anal. Non Linéaire, 7 (1990), 67-90. |
[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, J. Funct. Anal., 186 (2001), 432-520. |
[6] |
L. Bronsard and F. Reitich, On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation, Arch. Ration. Mech. Anal., 124 (1993), 355-379.
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, Comm. Pure Appl. Math., 47 (1994), 1457-1473. |
[8] |
L. Caffarelli and A. Córdoba, Uniform convergence of a singular perturbation problem, Comm. Pure Appl. Math., 48 (1995), 1-12.
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, J. Amer. Math. Soc., 21 (2008), 847-862.
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, Int. Math. Res. Not., 30 (2004), 1511-1527.
doi: 10.1155/S1073792804133588. |
[11] |
A. Farina, Two results on entire solutions of Ginzburg-Landau system in higher dimensions, J. Funct. Anal., 214 (2004), 386-395.
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, Arch. Ration. Mech. Anal., 195 (2010), 1025-1058.
doi: 10.1007/s00205-009-0227-8. |
[13] |
G. Fusco, On some elementary properties of vector minimizers of the Allen-Cahn energy, Comm. Pure Appl. Anal., 13 (2014), 1045-1060.
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, Comm. Partial Differential Equations, 39 (2014), 1032-1047.
doi: 10.1080/03605302.2013.849730. |
[15] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer-Verlag, New York, 1983. |
[16] |
C. Gui and F. Zhou, Asymptotic behavior of oscillating radial solutions to certain nonlinear equations, Methods Appl. Anal., 15 (2008), 285-296.
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 $\mathbbR^2$, Houston J. Math., 20 (1994), 653-669. |
[18] |
L. Modica, A gradient bound and a Liouville theorem for nonlinear Poisson equations, Comm. Pure Appl. Math., 38 (1985), 679-684.
doi: 10.1002/cpa.3160380515. |
[19] |
L. Modica, Monotonicity of the energy for entire solutions of semilinear elliptic equations, in Partial Differential Equations and the Calculus of Variations, Essays in honor of Ennio De Giorgi, Vol. 2 (eds. F. Colombini, A. Marino and L. Modica), Birkhäuser, (1989), 843-850. |
[20] |
T. Riviére, Asymptotic analysis for the Ginzburg-Landau equations, Bollettino dell'Unione Matematica Italiana 2-B, 3 (1999), 537-575. |
[21] |
D. Smets, PDE analysis of concentrating energies for the Ginzburg-Landau equation, in Topics on Concentration Phenomena and Problems with Multiple Scales (eds. A. Braides and V. C. Piat), Springer, (2006), 293-314.
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, New York, 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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