# American Institute of Mathematical Sciences

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] R. A. Adams, "Lecture Notes on $L^p$-Potential Theory,'' Department of Math., Univ. of Umeå, 1981. Google Scholar [2] N. Ansini and A. Braides, Asymptotic analysis of periodically-perforated nonlinear media, J. Math. Pures Appl. (9), 81 (2002), 439-451. doi: 10.1016/S0021-7824(01)01226-0.  Google Scholar [3] J. Bourgain, H. Brezis and P. Mironescu, Another look at Sobolev spaces, 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] A. Braides, "$\Gamma$-Convergence for Beginners,'' Oxford Lecture Series in Mathematics and its Applications, 22, Oxford University Press, Oxford, 2002.  Google Scholar [5] A. Braides and A. Defranceschi, "Homogenization of Multiple Integrals,'' Oxford Lecture Series in Mathematics and its Applications, 12, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar [6] H. Brezis, How to recognize constant functions. A connection with Sobolev spaces, Russian Math. Surveys, 57 (2002), 693-708. doi: 10.1070/RM2002v057n04ABEH000533.  Google Scholar [7] H. Brezis and L. Nirenberg, Degree theory and BMO. I. Compact manifolds without boundaries, Selecta Math. (N.S.), 1 (1995), 197-263.  Google Scholar [8] L. Caffarelli and A. Mellet, Random homogenization of an obstacle problem, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 375-395.  Google Scholar [9] L. Caffarelli and A. Mellet, Random homogenization of fractional obstacle problems, Netw. Heterog. Media, 3 (2008), 523-554. doi: 10.3934/nhm.2008.3.523.  Google Scholar [10] L. Caffarelli and L. Silvestre, An extension problem related to fractional Laplacians, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  Google Scholar [11] S. Conti, A. Garroni and S. Müller, Singular kernels, multiscale decomposition of microstructure, and dislocation models, Arch. Rational Mech. Anal., 199 (2011), 779-819. doi: 10.1007/s00205-010-0333-7.  Google Scholar [12] Ş. Costea, Strong $A_\infty$-weights and scaling invariant Besov capacities, Rev. Mat. Iberoam., 23 (2007), 1067-1114.  Google Scholar [13] G. Dal Maso, "An Introduction to $\Gamma$-Convergence,'' Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser Boston, Inc., Boston, MA, 1993.  Google Scholar [14] F. Daví and P. M. Mariano, Evolution of domain walls in ferroelectric solids, J. Mech. Phys. Solids, 49 (2001), 1701-1726. doi: 10.1016/S0022-5096(01)00014-X.  Google Scholar [15] M. Focardi, Homogenization of random fractional obstacle problems via $\Gamma$-convergence, Comm. Partial Differential Equations, 34 (2009), 1607-1631.  Google Scholar [16] M. Focardi, Aperiodic fractional obstacle problems, Adv. Math., 225 (2010), 3502-3544. doi: 10.1016/j.aim.2010.06.014.  Google Scholar [17] M. Focardi and A. Garroni, A $1D$ macroscopic phase field model for dislocations and a second order $\Gamma$-limit, Multiscale Model. Simul., 6 (2007), 1098-1124.  Google Scholar [18] A. Garroni and S. Müller, $\Gamma$-limit of a phase-field model of dislocations, SIAM J. Math. Anal., 36 (2005), 1943-1964.  Google Scholar [19] A. Garroni and S. Müller, A variational model for dislocations in the line tension limit, Arch. Ration. Mech. Anal., 181 (2006), 535-578. doi: 10.1007/s00205-006-0432-7.  Google Scholar [20] J. Heinonen, T. Kilpeläinen and O. Martio, "Nonlinear Potential Theory of Degenerate Elliptic Equations,'' Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993.  Google Scholar [21] M. Koslowski, A. M. Cuitiño and M. Ortiz, A phase-field theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystals, 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] M. Senechal, "Quasicrystals and Geometry,'' Cambridge University Press, Cambridge, 1995.  Google Scholar [24] L. Sigalotti, Asymptotic analysis of periodically-perforated nonlinear media at the critical exponent, Commun. Contemp. Math., 11 (2009), 1009-1033. doi: 10.1142/S0219199709003648.  Google Scholar [25] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,'' North-Holland Mathematical Library, 18, North Holland Publishing Co., Amsterdam-New York, 1978.  Google Scholar

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##### References:
 [1] R. A. Adams, "Lecture Notes on $L^p$-Potential Theory,'' Department of Math., Univ. of Umeå, 1981. Google Scholar [2] N. Ansini and A. Braides, Asymptotic analysis of periodically-perforated nonlinear media, J. Math. Pures Appl. (9), 81 (2002), 439-451. doi: 10.1016/S0021-7824(01)01226-0.  Google Scholar [3] J. Bourgain, H. Brezis and P. Mironescu, Another look at Sobolev spaces, 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] A. Braides, "$\Gamma$-Convergence for Beginners,'' Oxford Lecture Series in Mathematics and its Applications, 22, Oxford University Press, Oxford, 2002.  Google Scholar [5] A. Braides and A. Defranceschi, "Homogenization of Multiple Integrals,'' Oxford Lecture Series in Mathematics and its Applications, 12, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar [6] H. Brezis, How to recognize constant functions. A connection with Sobolev spaces, Russian Math. Surveys, 57 (2002), 693-708. doi: 10.1070/RM2002v057n04ABEH000533.  Google Scholar [7] H. Brezis and L. Nirenberg, Degree theory and BMO. I. Compact manifolds without boundaries, Selecta Math. (N.S.), 1 (1995), 197-263.  Google Scholar [8] L. Caffarelli and A. Mellet, Random homogenization of an obstacle problem, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 375-395.  Google Scholar [9] L. Caffarelli and A. Mellet, Random homogenization of fractional obstacle problems, Netw. Heterog. Media, 3 (2008), 523-554. doi: 10.3934/nhm.2008.3.523.  Google Scholar [10] L. Caffarelli and L. Silvestre, An extension problem related to fractional Laplacians, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  Google Scholar [11] S. Conti, A. Garroni and S. Müller, Singular kernels, multiscale decomposition of microstructure, and dislocation models, Arch. Rational Mech. Anal., 199 (2011), 779-819. doi: 10.1007/s00205-010-0333-7.  Google Scholar [12] Ş. Costea, Strong $A_\infty$-weights and scaling invariant Besov capacities, Rev. Mat. Iberoam., 23 (2007), 1067-1114.  Google Scholar [13] G. Dal Maso, "An Introduction to $\Gamma$-Convergence,'' Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser Boston, Inc., Boston, MA, 1993.  Google Scholar [14] F. Daví and P. M. Mariano, Evolution of domain walls in ferroelectric solids, J. Mech. Phys. Solids, 49 (2001), 1701-1726. doi: 10.1016/S0022-5096(01)00014-X.  Google Scholar [15] M. Focardi, Homogenization of random fractional obstacle problems via $\Gamma$-convergence, Comm. Partial Differential Equations, 34 (2009), 1607-1631.  Google Scholar [16] M. Focardi, Aperiodic fractional obstacle problems, Adv. Math., 225 (2010), 3502-3544. doi: 10.1016/j.aim.2010.06.014.  Google Scholar [17] M. Focardi and A. Garroni, A $1D$ macroscopic phase field model for dislocations and a second order $\Gamma$-limit, Multiscale Model. Simul., 6 (2007), 1098-1124.  Google Scholar [18] A. Garroni and S. Müller, $\Gamma$-limit of a phase-field model of dislocations, SIAM J. Math. Anal., 36 (2005), 1943-1964.  Google Scholar [19] A. Garroni and S. Müller, A variational model for dislocations in the line tension limit, Arch. Ration. Mech. Anal., 181 (2006), 535-578. doi: 10.1007/s00205-006-0432-7.  Google Scholar [20] J. Heinonen, T. Kilpeläinen and O. Martio, "Nonlinear Potential Theory of Degenerate Elliptic Equations,'' Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993.  Google Scholar [21] M. Koslowski, A. M. Cuitiño and M. Ortiz, A phase-field theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystals, 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] M. Senechal, "Quasicrystals and Geometry,'' Cambridge University Press, Cambridge, 1995.  Google Scholar [24] L. Sigalotti, Asymptotic analysis of periodically-perforated nonlinear media at the critical exponent, Commun. Contemp. Math., 11 (2009), 1009-1033. doi: 10.1142/S0219199709003648.  Google Scholar [25] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,'' North-Holland Mathematical Library, 18, North Holland Publishing Co., Amsterdam-New York, 1978.  Google Scholar
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