# American Institute of Mathematical Sciences

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

 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.
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

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##### 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
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