\`x^2+y_1+z_12^34\`
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.

    Citation:

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

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

    [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.

    [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.

    [4]

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

    [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.

    [6]

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

    [7]

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

    [8]

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

    [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.

    [10]

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

    [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.

    [12]

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

    [13]

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

    [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.

    [15]

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

    [16]

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

    [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.

    [18]

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

    [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.

    [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.

    [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.

    [22]

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

    [23]

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

    [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.

    [25]

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

  • 加载中
SHARE

Article Metrics

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

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return