Skip to main content
SpringerLink
Log in

The homogenization of surfaces and boundaries

  • Published:
Bulletin of the Brazilian Mathematical Society, New Series Aims and scope Submit manuscript

Abstract

In this survey I would like to describe several homogenization issues that arise when considering surfaces, embedded in a periodic media, that satisfy some associated geometrical property like minimizing a “periodic area integral,” or that are an interphase of a flow, a flame, a drop, whose shape is influenced, again by the surrounding periodic media.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. Aubry and P.Y. Le Daeron. The discrete Frenkel-Kontorova model and its extensions. I. Exact results for the ground-states. Phys. D, 8(3) (1983), 381–422.

    MATH  Google Scholar 

  2. F. Auer. Minimale hyperflächen in riemannsche n-torus. Albert-Ludwigs Univ. thesis (1997).

    Google Scholar 

  3. V. Bangert. On minimal laminationsof the torus. Ann. Inst. H. Poincaré Anal. Non Linéaire, 6(2) (1989), 95–138.

    MATH  MathSciNet  Google Scholar 

  4. V. Bangert. Minimalmeasures andminimizing closed normal one-currents. Geom. Funct. Anal., 9(3) (1999), 413–427.

    Article  MATH  MathSciNet  Google Scholar 

  5. L.A. Caffarelli and R. de la Llave. Planelikeminimizers in periodicmedia. Comm. Pure Appl. Math., 54(12) (2001), 1403–1441.

    Article  MATH  MathSciNet  Google Scholar 

  6. L.A. Caffarelli and R. de la Llave. Interfaces of ground states in Ising models with periodic coefficients. J. Stat. Phys., 118(3–4) (2005), 687–719.

    Article  MATH  MathSciNet  Google Scholar 

  7. A. Candel and R. de la Llave. On the Aubry-Mather theory in statisticalmechanics. Comm. Math. Phys., 192(3) (1998), 649–669.

    Article  MATH  MathSciNet  Google Scholar 

  8. A. Chambolle and G. Thouroude. Homogenizationof interfacial energies and construction of plane-likeminimizers in periodic media through a cell problem. Netw. Heterog. Media, 4(1) (2009), 127–152.

    Article  MATH  MathSciNet  Google Scholar 

  9. G.A. Hedlund. Geodesics on a two-dimensional Riemannian manifold with periodic coefficients. Ann. of Math., 33 (1932), 719–739.

    Article  MathSciNet  Google Scholar 

  10. M. Morse. A fundamental class of geodesics on any closed surface of genus greater than one. Trans. AMS., 26 (1924), 25–60.

    Article  MATH  MathSciNet  Google Scholar 

  11. J. Moser. Minimal solutions of variational problems on a torus. Ann. Inst. H. Poincaré Anal. Non Linéaire, 3(3) (1986), 229–272.

    MATH  Google Scholar 

  12. H. Berestycki and F. Hamel. Front propagation in periodic excitable media. Comm. Pure Appl. Math., 55(8) (2002), 949–1032.

    Article  MATH  MathSciNet  Google Scholar 

  13. L.A. Caffarelli and K.-A. Lee. Homogenizations of nonvariational viscosity solutions. Preprint (2005).

    Google Scholar 

  14. L.A. Caffarelli and K.-A. Lee. Homogenizationof the osciillating free boundaries: the elliptic case. Preprint (2005).

    Google Scholar 

  15. L.A. Caffarelli and A. Mellet. Capillary drops on an inhomogeneous surface: The horizontal plane. Preprint (2005).

    Google Scholar 

  16. L.A. Caffarelli, K.-A. Lee and A. Mellet. Singular limit and homogenization for flame propagation in periodic excitable media. Arch. Ration.Mech. Anal., 172(2) (2004), 153–190.

    Article  MATH  MathSciNet  Google Scholar 

  17. L.A. Caffarelli, K.-A. Lee and A. Mellet. Homogenization and flame propagation in periodic excitable media: The asymptotic speed of propagation. Comm. Pure Appl.Math., 59(4) (2006), 501–525.

    Article  MATH  MathSciNet  Google Scholar 

  18. L.A. Caffarelli, K.-A. Lee and A. Mellet. Flame propagation in one-dimensional stationary ergodic media. Math. Models Methods Appl. Sci., 17(1) (2007), 155–169.

    Article  MATH  MathSciNet  Google Scholar 

  19. L.A. Caffarelli and A. Mellet. Capillary drops: contact angle hysteresis and sticking drops. Calc. Var. Partial Differential Equations, 29(2) (2007), 141–160.

    Article  MATH  MathSciNet  Google Scholar 

  20. E. Gonzalez, U. Massari and I. Tamanini. On the regularity of boundaries of sets minimizing perimeter with a volume constraint. Indiana Univ. Math. J., 32(1) (1983), 25–37.

    Article  MATH  MathSciNet  Google Scholar 

  21. V. Dussan and R. Chow. On the ability of drops or bubbles to stick to nonhorizontal surfaces of solids. J. Fluid Mech., 137 (1983), 1–29.

    Article  MATH  Google Scholar 

  22. R.R. Hall. A quantitative isoperimetric inequality in n-dimensional space. J. Reine Angew. Math., 428 (1992), 161–176.

    MATH  MathSciNet  Google Scholar 

  23. L.M. Hocking. The wetting of a plane surface by a fluid. Phys. Fluids, 7(6) (1995), 1214–1220.

    Article  MATH  MathSciNet  Google Scholar 

  24. L.M. Hocking. On contact angles in evaporating liquids. Phys. Fluids, 7(12) (1995), 2950–2955.

    Article  MATH  MathSciNet  Google Scholar 

  25. C. Huh and S.G. Mason. Effects of Surface Roughness on Wetting (Theoretical). J. Colloid Interface Sci., 60 (1977), 11–38.

    Article  Google Scholar 

  26. J.-F. Joanny and P.-G. de Gennes. A model for contact angle hysteresis. J. Chem. Phys., 81 (1984), 552–562.

    Article  Google Scholar 

  27. L. Leger and J.-F. Joanny. Liquid Spreading. Rep. Prog. Phys., (1992), 431–486.

    Google Scholar 

  28. G. Barles. Some homogenization results for non-coercive Hamilton-Jacobi equations. Calculus of Variations and Partial Differential Equations, 30(4) (2007), 449–466.

    Article  MATH  MathSciNet  Google Scholar 

  29. G. Barles, A. Cesaroni and M. Novaga. Homogenizationof fronts in highly heterogeneous media. SIAM J. Math. Anal., 43(1) (2011), 212–227.

    Article  MATH  MathSciNet  Google Scholar 

  30. G. Barles, M. Soner and P.E. Souganidis. Front propagation and phase field theory. SIAM J. Control Optim., 31(2) (1993), 439–469.

    Article  MATH  MathSciNet  Google Scholar 

  31. L.A. Caffarelli and R. de la Llave. Planelikeminimizers in periodicmedia. Comm. Pure Appl. Math., 54(12) (2001), 1403–1441.

    Article  MATH  MathSciNet  Google Scholar 

  32. L.A. Caffarelli and R. Monneau. Counterexample in 3-D and homogenization in 2-D. Preprint HAL: hal-00720954 (version 1) (2012).

    Google Scholar 

  33. P. Cardaliaguet, P.-L. Lions and P.E. Souganidis. A discussion about the homogenization of moving interfaces. J. Math. Pures Appl. (9), 91(4) (2009), 339–363.

    Article  MATH  MathSciNet  Google Scholar 

  34. P. Cardaliaguet, J. Nolen and P.E. Souganidis. Homogenization and enhancement for the G-equation. Arch. Ration. Mech. Anal., 199(2) (2011), 527–561.

    Article  MATH  MathSciNet  Google Scholar 

  35. P. Cardaliaguet and P.E. Souganidis. Homogenization and enhancement of the Gequation in random environments. Preprint (2011).

    Google Scholar 

  36. A. Cesaroni and M. Novaga. Long-time behavior of the mean curvature flow with periodic forcing. Preprint (2011).

    Google Scholar 

  37. A. Cesaroni and M. Novaga. Homogenization of mean curvature flow of graphs in periodic media. Preprint (2012).

    Google Scholar 

  38. A. Cesaroni, M. Novaga and E. Valdinoci. Curvature flow in heterogeneousmedia. Preprint (2011).

    Google Scholar 

  39. B. Craciun and K. Bhattacharya. Effective motion of a curvature-sensitive interface through a heterogeneous medium. Interfaces Free Bound., 6(2) (2004), 151–173.

    Article  MATH  MathSciNet  Google Scholar 

  40. N. Dirr, G. Karali and N.K. Yip. Pulsating wave for mean curvature flow in inhomogeneous medium. European J. Appl. Math., 19(6) (2008), 661–699.

    Article  MATH  MathSciNet  Google Scholar 

  41. L.C. Evans and J. Spruck. Motion of level sets by mean curvature. I., J. Differential Geom., 33(3) (1991), 635–681.

    MATH  MathSciNet  Google Scholar 

  42. P.-L. Lions and P.E. Souganidis. Homogenizationof degenerate second-order PDE in periodic and almost periodic environments and applications. Ann. Inst. H. Poincaré Anal. Non Linéaire, 22(5) (2005), 667–677.

    Article  MATH  MathSciNet  Google Scholar 

  43. P.-L. Lions, G. Papanicolaou and S.R.S. Varadhan. Homogenization of Hamilton-Jacobi equations. Preprint.

  44. Y-Y. Liu, J. Xin and Y. Yu. Periodic homogenization of G-equations and viscosity effects. Nonlinearity, 23 (2010), 2351–2367.

    Article  MATH  MathSciNet  Google Scholar 

  45. B. Lou and X. Chen. Traveling waves of a curvature flow in almost periodicmedia. Differential Equations, 247(8) (2009), 2189–2208.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics — RLM 8.100, The University of Texas at Austin, 2515 Speedway — Stop C1200, Austin, TX, 78712-1202, USA

    Luis Caffarelli

Authors
  1. Luis Caffarelli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luis Caffarelli.

Additional information

Supported in part by NSF grant Nos. DMS-0654267 & DMS-1160802.

About this article

Cite this article

Caffarelli, L. The homogenization of surfaces and boundaries. Bull Braz Math Soc, New Series 44, 755–775 (2013). https://doi.org/10.1007/s00574-013-0033-7

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00574-013-0033-7

Keywords

Mathematical subject classification

Search

Navigation