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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 |
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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