Advanced Search
Article Contents
Article Contents

Vector-valued obstacle problems for non-local energies

Abstract Related Papers Cited by
  • 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.
    Mathematics Subject Classification: Primary: 74Q15, 35R11, 49J40.


    \begin{equation} \\ \end{equation}
  • [1]

    R. A. Adams, "Lecture Notes on $L^p$-Potential Theory,'' Department of Math., Univ. of Umeå, 1981.


    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.


    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.


    A. Braides, "$\Gamma$-Convergence for Beginners,'' Oxford Lecture Series in Mathematics and its Applications, 22, Oxford University Press, Oxford, 2002.


    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.


    H. Brezis, How to recognize constant functions. A connection with Sobolev spaces, Russian Math. Surveys, 57 (2002), 693-708.doi: 10.1070/RM2002v057n04ABEH000533.


    H. Brezis and L. Nirenberg, Degree theory and BMO. I. Compact manifolds without boundaries, Selecta Math. (N.S.), 1 (1995), 197-263.


    L. Caffarelli and A. Mellet, Random homogenization of an obstacle problem, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 375-395.


    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.


    L. Caffarelli and L. Silvestre, An extension problem related to fractional Laplacians, Comm. Partial Differential Equations, 32 (2007), 1245-1260.


    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.


    Ş. Costea, Strong $A_\infty$-weights and scaling invariant Besov capacities, Rev. Mat. Iberoam., 23 (2007), 1067-1114.


    G. Dal Maso, "An Introduction to $\Gamma$-Convergence,'' Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser Boston, Inc., Boston, MA, 1993.


    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.


    M. Focardi, Homogenization of random fractional obstacle problems via $\Gamma$-convergence, Comm. Partial Differential Equations, 34 (2009), 1607-1631.


    M. Focardi, Aperiodic fractional obstacle problems, Adv. Math., 225 (2010), 3502-3544.doi: 10.1016/j.aim.2010.06.014.


    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.


    A. Garroni and S. Müller, $\Gamma$-limit of a phase-field model of dislocations, SIAM J. Math. Anal., 36 (2005), 1943-1964.


    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.


    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.


    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.


    O. Savin and E. Valdinoci$\Gamma$-convergence for nonlocal phase transitions, preprint, arXiv:1007.1725.


    M. Senechal, "Quasicrystals and Geometry,'' Cambridge University Press, Cambridge, 1995.


    L. Sigalotti, Asymptotic analysis of periodically-perforated nonlinear media at the critical exponent, Commun. Contemp. Math., 11 (2009), 1009-1033.doi: 10.1142/S0219199709003648.


    H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,'' North-Holland Mathematical Library, 18, North Holland Publishing Co., Amsterdam-New York, 1978.

  • 加载中

Article Metrics

HTML views() PDF downloads(126) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint