June  2020, 40(6): 3253-3290. doi: 10.3934/dcds.2020124

Boundary spike of the singular limit of an energy minimizing problem

Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA 15260, USA

Received  February 2019 Revised  September 2019 Published  February 2020

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

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.

Citation: Xinfu Chen, Huiqiang Jiang, Guoqing Liu. Boundary spike of the singular limit of an energy minimizing problem. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3253-3290. doi: 10.3934/dcds.2020124
References:
[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. L. Bertozzi, The mathematics of moving contact lines in thin liquid films, Notices Amer. Math. Soc., 45 (1998), 689-697.   Google Scholar

[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.   Google Scholar

[12]

A. Boulkhemair and A. Chakib, On the uniform Poincaré inequality, Comm. Partial Differential Equations, 32 (2007), 1439-1447.  doi: 10.1080/03605300600910241.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

H. Jiang and A. Miloua, Point rupture solutions of a singular elliptic equation, Electron. J. Differential Equations, 2013 (2013), pages No. 70, 8pp.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[29]

C. Lederman, A free boundary problem with a volume penalization, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 23 (1996), 249–300.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[33]

D. Slepčev, Linear stability of selfsimilar solutions of unstable thin-film equations, Interfaces Free Bound., 11 (2009), 375-398.   Google Scholar

[34]

G. S. Weiss, Partial regularity for a minimum problem with free boundary, J. Geom. Anal., 9 (1999), 317-326.  doi: 10.1007/BF02921941.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. L. Bertozzi, The mathematics of moving contact lines in thin liquid films, Notices Amer. Math. Soc., 45 (1998), 689-697.   Google Scholar

[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.   Google Scholar

[12]

A. Boulkhemair and A. Chakib, On the uniform Poincaré inequality, Comm. Partial Differential Equations, 32 (2007), 1439-1447.  doi: 10.1080/03605300600910241.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

H. Jiang and A. Miloua, Point rupture solutions of a singular elliptic equation, Electron. J. Differential Equations, 2013 (2013), pages No. 70, 8pp.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[29]

C. Lederman, A free boundary problem with a volume penalization, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 23 (1996), 249–300.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[33]

D. Slepčev, Linear stability of selfsimilar solutions of unstable thin-film equations, Interfaces Free Bound., 11 (2009), 375-398.   Google Scholar

[34]

G. S. Weiss, Partial regularity for a minimum problem with free boundary, J. Geom. Anal., 9 (1999), 317-326.  doi: 10.1007/BF02921941.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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