On some elementary properties of vector minimizers of the Allen-Cahn energy

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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

Giorgio Fusco ^1,

1.	Dipartimento di Matematica Pura e Applicata, Università de L’Aquila, I-67100 L’Aquila

Received March 2013 Revised September 2013 Published December 2013

Abstract
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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.

Keywords: point-wise estimate., Local minimizer, polar representation, deformation lemma, energy estimates.

Mathematics Subject Classification: Primary: 35J47, 35J50; Secondary: 35J2.

Citation: Giorgio Fusco. On some elementary properties of vector minimizers of the Allen-Cahn energy. Communications on Pure and 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, Acta Appl. Math., 65 (2001), 9-33. doi: 10.1023/A:1010602715526.
[2]	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.
[3]	N. D. Alikakos and G. Fusco, Entire solutions to equivariant elliptic system with variational structure, Arch. Rational Mech. Anal., 202 (2011), 567-597. doi: 10.1007/s00205-011-0441-z.
[4]	N. D. Alikakos and G. Fusco, Asymptotic rigidity results for symmetric solutions of the elliptic system $\Delta u = Wu(u)$,, work in progress., ().
[5]	N. D. Alikakos and G. Fusco, A maximum principle for systems with variational structure and an application to standing waves, preprint, (2012).
[6]	P. W. Bates, G. Fusco and P. Smyrnelis, Entire solutions with six-fold junctions to elliptic gradient systems with triangle symmetry, Advan. Nonlin. Stud., 13 (2013), 1-13.
[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., ().
[8]	A. Czarnecki, M. Kulczychi and W. Lubawski, On the connectedness of boundary and complement for domains, Ann. Polin. Math., 103 (2011), 189-191. doi: 10.4064/ap103-2-6.
[9]	G. Fusco, Equivariant entire solutions to the elliptic system $\Delta u=W_u(u)$ for general $G-$invariant potentials, Calc. Var. Part. Diff. Eqs., (2013), 1-23.
[10]	G. Fusco, F. Leonetti and C. Pignotti, A uniform estimate for positive solutions of semilinear elliptic equations, Trans. Amer. Math. Soc., 363 (2011), 4285-4307. doi: 10.1090/S0002-9947-2011-05356-0.
[11]	B. Gidas, W. M. Ni and L. Niremberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. doi: 10.1007/BF01221125.
[12]	J. Liouville, Lecons sur les fonctions doublement pèriodiques, J. Reine Angew. Math., 88 (1879), 277-310.

show all references

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, Acta Appl. Math., 65 (2001), 9-33. doi: 10.1023/A:1010602715526.
[2]	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.
[3]	N. D. Alikakos and G. Fusco, Entire solutions to equivariant elliptic system with variational structure, Arch. Rational Mech. Anal., 202 (2011), 567-597. doi: 10.1007/s00205-011-0441-z.
[4]	N. D. Alikakos and G. Fusco, Asymptotic rigidity results for symmetric solutions of the elliptic system $\Delta u = Wu(u)$,, work in progress., ().
[5]	N. D. Alikakos and G. Fusco, A maximum principle for systems with variational structure and an application to standing waves, preprint, (2012).
[6]	P. W. Bates, G. Fusco and P. Smyrnelis, Entire solutions with six-fold junctions to elliptic gradient systems with triangle symmetry, Advan. Nonlin. Stud., 13 (2013), 1-13.
[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., ().
[8]	A. Czarnecki, M. Kulczychi and W. Lubawski, On the connectedness of boundary and complement for domains, Ann. Polin. Math., 103 (2011), 189-191. doi: 10.4064/ap103-2-6.
[9]	G. Fusco, Equivariant entire solutions to the elliptic system $\Delta u=W_u(u)$ for general $G-$invariant potentials, Calc. Var. Part. Diff. Eqs., (2013), 1-23.
[10]	G. Fusco, F. Leonetti and C. Pignotti, A uniform estimate for positive solutions of semilinear elliptic equations, Trans. Amer. Math. Soc., 363 (2011), 4285-4307. doi: 10.1090/S0002-9947-2011-05356-0.
[11]	B. Gidas, W. M. Ni and L. Niremberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. doi: 10.1007/BF01221125.
[12]	J. Liouville, Lecons sur les fonctions doublement pèriodiques, J. Reine Angew. Math., 88 (1879), 277-310.

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