March  2012, 17(2): 487-507. doi: 10.3934/dcdsb.2012.17.487

Vector-valued obstacle problems for non-local energies

1. 

Dip. Mat. “U. Dini”, Università di Firenze, V.le Morgagni 67/A, I-50134 Firenze

Received  November 2010 Revised  March 2011 Published  December 2011

We investigate the asymptotics of obstacle problems for non-local energies in a vector-valued setting. Motivations arise, in particular, in phase field models for ferroelectric materials and variational theories for dislocations.
Citation: Matteo Focardi. Vector-valued obstacle problems for non-local energies. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 487-507. doi: 10.3934/dcdsb.2012.17.487
References:
[1]

Department of Math., Univ. of Umeå, 1981. Google Scholar

[2]

J. Math. Pures Appl. (9), 81 (2002), 439-451. doi: 10.1016/S0021-7824(01)01226-0.  Google Scholar

[3]

in "Optimal Control and Partial Differential Equations" (eds. J.L. Menaldi, E. Rofman and A. Sulem), A Volume in Honour of A. Bensoussan's 60th Birthday, IOS Press, (2001), 439-455. Google Scholar

[4]

Oxford Lecture Series in Mathematics and its Applications, 22, Oxford University Press, Oxford, 2002.  Google Scholar

[5]

Oxford Lecture Series in Mathematics and its Applications, 12, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[6]

Russian Math. Surveys, 57 (2002), 693-708. doi: 10.1070/RM2002v057n04ABEH000533.  Google Scholar

[7]

Selecta Math. (N.S.), 1 (1995), 197-263.  Google Scholar

[8]

Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 375-395.  Google Scholar

[9]

Netw. Heterog. Media, 3 (2008), 523-554. doi: 10.3934/nhm.2008.3.523.  Google Scholar

[10]

Comm. Partial Differential Equations, 32 (2007), 1245-1260.  Google Scholar

[11]

Arch. Rational Mech. Anal., 199 (2011), 779-819. doi: 10.1007/s00205-010-0333-7.  Google Scholar

[12]

Rev. Mat. Iberoam., 23 (2007), 1067-1114.  Google Scholar

[13]

Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser Boston, Inc., Boston, MA, 1993.  Google Scholar

[14]

J. Mech. Phys. Solids, 49 (2001), 1701-1726. doi: 10.1016/S0022-5096(01)00014-X.  Google Scholar

[15]

Comm. Partial Differential Equations, 34 (2009), 1607-1631.  Google Scholar

[16]

Adv. Math., 225 (2010), 3502-3544. doi: 10.1016/j.aim.2010.06.014.  Google Scholar

[17]

Multiscale Model. Simul., 6 (2007), 1098-1124.  Google Scholar

[18]

SIAM J. Math. Anal., 36 (2005), 1943-1964.  Google Scholar

[19]

Arch. Ration. Mech. Anal., 181 (2006), 535-578. doi: 10.1007/s00205-006-0432-7.  Google Scholar

[20]

Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993.  Google Scholar

[21]

J. Mech. Phys. Solids, 50 (2002), 2597-2635. doi: 10.1016/S0022-5096(02)00037-6.  Google Scholar

[22]

O. Savin and E. Valdinoci, $\Gamma$-convergence for nonlocal phase transitions,, preprint, ().   Google Scholar

[23]

Cambridge University Press, Cambridge, 1995.  Google Scholar

[24]

Commun. Contemp. Math., 11 (2009), 1009-1033. doi: 10.1142/S0219199709003648.  Google Scholar

[25]

North-Holland Mathematical Library, 18, North Holland Publishing Co., Amsterdam-New York, 1978.  Google Scholar

show all references

References:
[1]

Department of Math., Univ. of Umeå, 1981. Google Scholar

[2]

J. Math. Pures Appl. (9), 81 (2002), 439-451. doi: 10.1016/S0021-7824(01)01226-0.  Google Scholar

[3]

in "Optimal Control and Partial Differential Equations" (eds. J.L. Menaldi, E. Rofman and A. Sulem), A Volume in Honour of A. Bensoussan's 60th Birthday, IOS Press, (2001), 439-455. Google Scholar

[4]

Oxford Lecture Series in Mathematics and its Applications, 22, Oxford University Press, Oxford, 2002.  Google Scholar

[5]

Oxford Lecture Series in Mathematics and its Applications, 12, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[6]

Russian Math. Surveys, 57 (2002), 693-708. doi: 10.1070/RM2002v057n04ABEH000533.  Google Scholar

[7]

Selecta Math. (N.S.), 1 (1995), 197-263.  Google Scholar

[8]

Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 375-395.  Google Scholar

[9]

Netw. Heterog. Media, 3 (2008), 523-554. doi: 10.3934/nhm.2008.3.523.  Google Scholar

[10]

Comm. Partial Differential Equations, 32 (2007), 1245-1260.  Google Scholar

[11]

Arch. Rational Mech. Anal., 199 (2011), 779-819. doi: 10.1007/s00205-010-0333-7.  Google Scholar

[12]

Rev. Mat. Iberoam., 23 (2007), 1067-1114.  Google Scholar

[13]

Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser Boston, Inc., Boston, MA, 1993.  Google Scholar

[14]

J. Mech. Phys. Solids, 49 (2001), 1701-1726. doi: 10.1016/S0022-5096(01)00014-X.  Google Scholar

[15]

Comm. Partial Differential Equations, 34 (2009), 1607-1631.  Google Scholar

[16]

Adv. Math., 225 (2010), 3502-3544. doi: 10.1016/j.aim.2010.06.014.  Google Scholar

[17]

Multiscale Model. Simul., 6 (2007), 1098-1124.  Google Scholar

[18]

SIAM J. Math. Anal., 36 (2005), 1943-1964.  Google Scholar

[19]

Arch. Ration. Mech. Anal., 181 (2006), 535-578. doi: 10.1007/s00205-006-0432-7.  Google Scholar

[20]

Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993.  Google Scholar

[21]

J. Mech. Phys. Solids, 50 (2002), 2597-2635. doi: 10.1016/S0022-5096(02)00037-6.  Google Scholar

[22]

O. Savin and E. Valdinoci, $\Gamma$-convergence for nonlocal phase transitions,, preprint, ().   Google Scholar

[23]

Cambridge University Press, Cambridge, 1995.  Google Scholar

[24]

Commun. Contemp. Math., 11 (2009), 1009-1033. doi: 10.1142/S0219199709003648.  Google Scholar

[25]

North-Holland Mathematical Library, 18, North Holland Publishing Co., Amsterdam-New York, 1978.  Google Scholar

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