Advanced Search
Article Contents
Article Contents

Boundary spike of the singular limit of an energy minimizing problem

This research is partially supported by the National Science Foundation grants DMS-0504691 and DMS-1200599

Abstract Full Text(HTML) Related Papers Cited by
  • In this paper, we consider the singular limit of an energy minimizing problem which is a semi-limit of a singular elliptic equation modeling steady states of thin film equation with both Van der Waals force and Born repulsion force. We show that the singular limit of energy minimizers is a Dirac mass located on the boundary point with the maximum curvature.

    Mathematics Subject Classification: Primary: 35J20, 74K35; Secondary: 34B18.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] N. AguileraH. W. Alt and L. A. Caffarelli, An optimization problem with volume constraint, SIAM J. Control Optim., 24 (1986), 191-198.  doi: 10.1137/0324011.
    [2] H. W. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math., 325 (1981), 105-144. 
    [3] H. W. Alt, L. A. Caffarelli and A. Friedman, A free boundary problem for quasilinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 11 (1984), 1–44.
    [4] H. W. AltL. A. Caffarelli and A. Friedman, Variational problems with two phases and their free boundaries, Trans. Amer. Math. Soc., 282 (1984), 431-461.  doi: 10.1090/S0002-9947-1984-0732100-6.
    [5] S. Axler, P. Bourdon and W. Ramey, Harmonic Function Theory, volume 137 of Graduate Texts in Mathematics, Springer-Verlag, New York, second edition, 2001. doi: 10.1007/978-1-4757-8137-3.
    [6] G. I. BarenblattE. Beretta and M. Bertsch, The problem of the spreading of a liquid film along a solid surface: A new mathematical formulation, Proc. Nat. Acad. Sci. U.S.A., 94 (1997), 10024-10030.  doi: 10.1073/pnas.94.19.10024.
    [7] E. Beretta, Selfsimilar source solutions of a fourth order degenerate parabolic equation, Nonlinear Anal., 29 (1997), 741-760.  doi: 10.1016/S0362-546X(97)81321-1.
    [8] A. L. BertozziG. Grün and T. P. Witelski, Dewetting films: Bifurcations and concentrations, Nonlinearity, 14 (2001), 1569-1592.  doi: 10.1088/0951-7715/14/6/309.
    [9] A. L. Bertozzi and M. Pugh, The lubrication approximation for thin viscous films: Regularity and long-time behavior of weak solutions, Comm. Pure Appl. Math., 49 (1996), 85-123.  doi: 10.1002/(SICI)1097-0312(199602)49:2<85::AID-CPA1>3.0.CO;2-2.
    [10] A. L. Bertozzi, The mathematics of moving contact lines in thin liquid films, Notices Amer. Math. Soc., 45 (1998), 689-697. 
    [11] M. BertschR. Dal PassoH. Garcke and G. Grün, The thin viscous flow equation in higher space dimensions, Adv. Differential Equations, 3 (1998), 417-440. 
    [12] A. Boulkhemair and A. Chakib, On the uniform Poincaré inequality, Comm. Partial Differential Equations, 32 (2007), 1439-1447.  doi: 10.1080/03605300600910241.
    [13] L. A. Caffarelli, D. Jerison and C. E. Kenig, Global energy minimizers for free boundary problems and full regularity in three dimensions, In Noncompact Problems at the Intersection of Geometry, Analysis, and Topology, volume 350 of Contemp. Math., pages 83–97. Amer. Math. Soc., Providence, RI, 2004. doi: 10.1090/conm/350/06339.
    [14] L. A. CaffarelliJ.-M. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc. (JEMS), 12 (2010), 1151-1179.  doi: 10.4171/JEMS/226.
    [15] X. Chen and H. Jiang, Singular limit of an energy minimizer arising from dewetting thin film model with van der Waal, Born repulsion and surface tension forces, Calc. Var. Partial Differential Equations, 44 (2012), 221-246.  doi: 10.1007/s00526-011-0432-9.
    [16] D. Danielli and A. Petrosyan, A minimum problem with free boundary for a degenerate quasilinear operator, Calc. Var. Partial Differential Equations, 23 (2005), 97-124.  doi: 10.1007/s00526-004-0294-5.
    [17] H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969.
    [18] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, volume 105 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1983.
    [19] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations Of Second Order, Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.
    [20] D. Jerison and O. Savin, Some remarks on stability of cones for the one-phase free boundary problem, Geometric and Functional Analysis, 25 (2015), 1240-1257.  doi: 10.1007/s00039-015-0335-6.
    [21] H. Jiang, Energy minimizers of a thin film equation with Born repulsion force, Commun. Pure Appl. Anal., 10 (2011), 803-815.  doi: 10.3934/cpaa.2011.10.803.
    [22] H. Jiang and F. Lin, Zero set of Sobolev functions with negative power of integrability, Chinese Ann. Math. Ser. B, 25 (2004), 65-72.  doi: 10.1142/S0252959904000068.
    [23] H. Jiang and A. Miloua, Point rupture solutions of a singular elliptic equation, Electron. J. Differential Equations, 2013 (2013), pages No. 70, 8pp.
    [24] H. Jiang and W.-M. Ni, On steady states of van der Waals force driven thin film equations, European J. Appl. Math., 18 (2007), 153-180.  doi: 10.1017/S0956792507006936.
    [25] R. S. Laugesen and M. C. Pugh, Linear stability of steady states for thin film and Cahn-Hilliard type equations, Arch. Ration. Mech. Anal., 154 (2000), 3-51.  doi: 10.1007/PL00004234.
    [26] R. S. Laugesen and M. C. Pugh, Properties of steady states for thin film equations, European J. Appl. Math., 11 (2000), 293-351.  doi: 10.1017/S0956792599003794.
    [27] R. S. Laugesen and M. C. Pugh, Energy levels of steady states for thin-film-type equations, J. Differential Equations, 182 (2002), 377-415.  doi: 10.1006/jdeq.2001.4108.
    [28] R. S. Laugesen and M. C. Pugh, Heteroclinic orbits, mobility parameters and stability for thin film type equations, Electron. J. Differential Equations, 2002 (2002), pages No. 95, 29pp.
    [29] C. Lederman, A free boundary problem with a volume penalization, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 23 (1996), 249–300.
    [30] W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math., 44 (1991), 819-851.  doi: 10.1002/cpa.3160440705.
    [31] W.-M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J., 70 (1993), 247-281. 
    [32] W.-M. Ni and J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math., 48 (1995), 731-768.  doi: 10.1002/cpa.3160480704.
    [33] D. Slepčev, Linear stability of selfsimilar solutions of unstable thin-film equations, Interfaces Free Bound., 11 (2009), 375-398. 
    [34] G. S. Weiss, Partial regularity for a minimum problem with free boundary, J. Geom. Anal., 9 (1999), 317-326.  doi: 10.1007/BF02921941.
    [35] T. P. Witelski and A. J. Bernoff, Stability of self-similar solutions for van der Waals driven thin film rupture, Phys. Fluids, 11 (1999), 2443-2445.  doi: 10.1063/1.870138.
    [36] T. P. Witelski and A. J. Bernoff, Dynamics of three-dimensional thin film rupture, Phys. D, 147 (2000), 155-176.  doi: 10.1016/S0167-2789(00)00165-2.
  • 加载中
Open Access Under a Creative Commons license

Article Metrics

HTML views(2081) PDF downloads(373) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint