Plane-like minimal surfaces in periodic media with exclusions

SIAM Journal on Mathematical Analysis

Volumn 36, Issue 2, 2005, Pages 523-551

  Torres, Monica ^a  

^a NORTHWESTERN UNIVERSITY   (United States)

Author keywords

Homogenization; Minimal surfaces; Periodic media; Sets of finite perimeter

Indexed keywords

GEOMETRIC MEASURE THEORY; HOMOGENIZATION; MINIMAL SURFACES; PERIODIC MEDIA; PLANE-LIKE MINIMAL SURFACES; SETS OF FINITE PERIMETER;

BOUNDARY CONDITIONS; CRYSTAL LATTICES; FUNCTIONS; HAMILTONIANS; INTEGRAL EQUATIONS; MATHEMATICAL MODELS; NUMERICAL ANALYSIS; PERTURBATION TECHNIQUES; PROBLEM SOLVING; SET THEORY; THEOREM PROVING; TIME VARYING SYSTEMS;

SURFACE PHENOMENA;

EID: 14944367390     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036141001399970     Document Type: Article

References (36)

1. O. ALVAREZ, Homogenization of Hamilton-Jacobi equations in perforated sets, J. Differential Equations, 159 (1999), pp. 543-577.
2. L. AMBROSIO, N. FUSCO, AND D. PALLARA, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000.
3. F. AUER, Minimale hyperflächen in riemannsche n-torus, Albert-Ludwigs Univ. thesis, Freiburg, Germany, 1997.
4. F. AUER AND V. BANGERT, Minimising currents and the stable norm in codimension one, C. R. Acad. Sci. Paris Sér. I Math., 333 (2001), pp. 1095-1100.
5. V. BANGERT, Minimal geodesics, Ergodic Theory Dynam. Systems, 10 (1990), pp. 263-286.
6. V. BANGERT, Minimal foliations and laminations, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Birkhäuser, Basel, 1995, pp. 453-464.
7. M. BARDI, M. G. CRANDALL, L. C. EVANS, H. M. SONER, AND P. E. SOUGANIDIS, Viscosity Solutions and Applications, Lecture Notes in Math. 1660, I. Capuzzo Dolcetta and P. L. Lions, eds., Springer-Verlag, Berlin, 1997.
8. M. BARDI AND I. CAPUZZO-DOLCETTA, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser Boston, Boston, MA, 1997.
9. G. BARLES AND P. E. SOUGANIDIS, On the large time behavior of solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal., 31 (2000), pp. 925-939.
10. D. BURAGO, Periodic metrics, in Seminar on Dynamical Systems (St. Petersburg, 1991), Progr. Nonlinear Differential Equations Appl. 12, Birkhäuser, Basel, 1994, pp. 90-95.
11. D. BURAGO, S. IVANOV, AND B. KLEINER, On the structure of the stable norm of periodic metrics, Math. Res. Lett., 4 (1997), pp. 791-808.
12. L. A. CAFFARELLI AND A. CÓRDOBA, An elementary regularity theory of minimal surfaces, Differential Integral Equations, 6 (1993), pp. 1-13.
13. L. A. CAFFARELLI AND A. CÓRDOBA, Correction: "An elementary regularity theory of minimal surfaces" [Differential Integral Equations, 6 (1993), pp. 1-13], Differential Integral Equations, 8 (1995), p. 223.
14. L. A. CAFFARELLI AND A. CÓRDOBA, Correction: "An elementary regularity theory of minimal surfaces" [Differential Integral Equations, 6 (1993), pp. 1-13], Differential Integral Equations, 8 (1995), p. 223.
15. L. A. CAFFARELLI AND R. DE LA LLAVE, Planelike minimizers in periodic media, Comm. Pure Appl. Math., 54 (2001), pp. 1403-1441,
16. I. CAPUZZO-DOLCETTA AND H, ISHII, On the rate of convergence in homogenization of Hamilton-Jacobi equations, Indiana Univ. Math. J., 50 (2001), pp. 1113-1129.
17. M. C. CONCORDEL, Periodic homogenization of Hamilton- Jacobi equations: Additive eigenvalues and variational formula, Indiana Univ. Math. J., 45 (1996), pp. 1095-1117.
18. M. C. CONCORDEL, Periodic homogenisation of Hamilton-Jacobi equations. II. Eikonal equations, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), pp. 665-689.
19. E. DE GIORGI, Frontiere orientate di misura minima, Seminario di Matematica della Scuola Normale Superiore di Pisa, 1960-61, Editrice Tecnico Scientifica, Pisa, 1961.
20. L. C. EVANS AND D. GOMES, Effective Hamiltonians and averaging for Hamiltonian dynamics. I, Arch. Ration. Mech. Anal., 157 (2001), pp. 1-33.
21. L. C. EVANS AND D. GOMES, Effective Hamiltonians and averaging for Hamiltonian dynamics. II, Arch. Ration. Mech. Anal., 161 (2002), pp. 271-305.
22. L. C. EVANS, The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), pp. 359-375.
23. L. C. EVANS, Periodic homogenisation of certain fully nonlinear partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), pp. 245-265.
24. L. C. EVANS, Partial differential equations, Graduate Stud. Math. 19, AMS, Providence, RI, 1998.
25. L. C. EVANS, Effective Hamiltonians and quantum states, in Séminaire: Équations aux Dérivées Partielles, 2000-2001, Sémin. Équ. Dériv. Partielles, Exp. No. XXII, École Polytech., Palaiseau, 2001.
26. L. C. EVANS AND R. F. GARIEPY, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, FL, 1992.
27. H. FEDERER, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag, New York, 1969.
28. E. GIUSTI, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser Verlag, Basel, 1984.
29. M. GOLDBERG, Stable norms - examples and remarks, in General inequalities, 7 (Oberwolfach, 1995), Internat. Ser. Numer. Math. 123, Birkhäuser, Basel, 1997, pp. 61-64.
30. M. GOLDBERG, Stable norms - examples and remarks, in General inequalities, 7 (Oberwolfach, 1995), Internat. Ser. Numer. Math. 123, Birkhäuser, Basel, 1997, pp. 61-64.
31. M. GRÜTER, Optimal regularity for codimension one minimal surfaces with a free boundary, Manuscripta Math., 58 (1987), pp. 295-343.
32. K. HORIE AND H. ISHII, Homogenization of Hamilton-Jacobi equations on domains with small scale periodic structure, Indiana Univ. Math. J., 47 (1998), pp. 1011-1058.
33. Z. LIN, L. XIAO-BIAO, M. TORRES, AND Z. HONGKAI, Theoretical and numerical analysis of the weighted minimal surface problem, Methods Appl. Anal., 10 (2003), pp. 199-214.
34. P. L. LIONS, G. PAPANICOLAOU, AND S. R. S. VARADHAN, Homogenization of Hamilton-Jacobi Equations, manuscript.
35. D. MASSART, Stable norms of surfaces: Local structure of the unit ball of rational directions, Geom. Funct. Anal., 7 (1997), pp. 996-1010.
36. L. SIMON, Lectures on Geometric Measure Theory, Australian National University, Centre for Mathematical Analysis, Canberra, 1983.