Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Surfaces minimizing area plus length of singular curves


Author: Frank Morgan
Journal: Proc. Amer. Math. Soc. 122 (1994), 1153-1161
MSC: Primary 49Q05; Secondary 49Q20
DOI: https://doi.org/10.1090/S0002-9939-1994-1231039-1
MathSciNet review: 1231039
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give the first existence and regularity results on surfaces minimizing area plus length of singular curves, as in energy-minimizing interfaces in materials.


References [Enhancements On Off] (What's this?)

  • [Ab] Jeff Abrahamson, Curves length minimizing modulo $ \nu$ in $ {{\mathbf{R}}^n}$, Michigan Math. J. 35 (1988), 285-290. MR 959275 (89h:49036)
  • [All1] William K. Allard, On boundary regularity for Plateau's problem, Bull Amer. Math. Soc. 75 (1969), 522-523. MR 0241802 (39:3140)
  • [All2] -, On the first variation of a varifold: Boundary behavior, Ann. of Math. (2) 101 (1975), 418-446. MR 0397520 (53:1379)
  • [Alm] F. J. Almgren, Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. Amer. Math. Soc., vol. 4, No. 165, Amer. Math. Soc., Providence, RI, 1976. MR 0420406 (54:8420)
  • [Alt] Hans Wilhelm Alt, Die Existenz eines Minimalfläche mit freimem Rand vorgeschriebrener Länge, Arch. Rational Mech. Anal. 51 (1973), 304-320. MR 0336528 (49:1302)
  • [Be] Scott Berger, The search for boundary-minimizing polytopes in two and three dimensions, Junior Paper, Princeton University, Princeton, NJ, June 1991.
  • [Br1] Kenneth A. Brakke, Minimal cones on hypercubes, J. Geom. Anal. 1 (1991), 329-338. MR 1129346 (92k:49082)
  • [Br2] -, The surface evolver, Experiment. Math. 1 (1992), 141-165. MR 1203871 (93k:53006)
  • [CCF] T. Colthurst, C. Cox, J. Foisy, H. Howards, K. Kollett, H. Lowy, and S. Root, Networks minimizing length plus the number of Steiner points, Williams College SMALL Undergraduate Research Geometry Group, Network Optimization Problems: Algorithms, Complexity and Applications (Dingzhu Du and Panos M. Pardalos, eds.), World Scientific, 1993, pp. 23-26.
  • [C] Edith A. Cook, Free boundary regularity for surfaces minimizing $ {\text{Area}}(S) + c\;{\text{area}}(\partial S)$, Trans. Amer. Math. Soc. 290 (1985), 503-526. MR 792809 (87h:49046)
  • [DG] E. De Giorgi, Su alcuni problemi comuni all'analisi e alla geometria, Note Mat. IX-Suppl. (1989), 59-71.
  • [E] Klaus Ecker, Area-minimizing integral currents with movable boundary parts of prescribed mass, Ann. Inst. H. Poincaré 6 (1989), 261-293. MR 998604 (90f:49036)
  • [F1] Herbert Federer, Geometric measure theory, Springer-Verlag, New York, 1969. MR 0257325 (41:1976)
  • [F2] -, The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension, Bull. Amer. Math. Soc. 76 (1970), 767-771. MR 0260981 (41:5601)
  • [FT] L. Fejes-Toth, Regular figures, Macmillan, New York, 1964. MR 0165423 (29:2705)
  • [GT] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, New York, 1983. MR 737190 (86c:35035)
  • [H1] S. Hildebrandt, Ein einfacher Beweis für die Regularität der Lösungen gewisser zwei dimensionaler Variations probleme unter freien Randbedingungen, Math. Ann. 194 (1971), 316-331. MR 0291927 (45:1015)
  • [H2] -, On two isoperimetric problems with free boundary conditions, Variational Methods for Free Surface Interfaces (Paul Concus and Robert Finn, eds.), Springer-Verlag, New York, 1986, pp. 43-51. MR 872887
  • [LM] Gary Lawlor and Frank Morgan, Paired calibrations applied to soap films, immiscible fluids, and surfaces or networks minimizing other norms, Pacific J. Math. (to appear). MR 1306034 (95i:58051)
  • [Me] Z. A. Melzak, Problems connected with convexity, Canad. Math. J. 8 (1965), 565-573. MR 0196594 (33:4781)
  • [M1] Frank Morgan, Clusters minimizing area plus length of singular curves, Math. Ann. (to appear). MR 1286892 (95g:49083)
  • [M2] -, Geometric measure theory: a beginner's guide, Academic Press, New York, 1988. MR 933756 (89f:49036)
  • [MT] Frank Morgan and Jean Taylor, Destabilization of the tetrahedral point junction by positive triple junction line energy, Scripta Met. Mat. 25 (1991), 1907-1910.
  • [MuS] David Mumford and Jayant Shah, Optimal approximation by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math. 42 (1989), 577-685. MR 997568 (90g:49033)
  • [N] Johannes C. C. Nitsche, The regularity of minimal surfaces on the movable parts of their boundaries, Indiana Univ. Math. J. 21 (1971), 505-513. MR 0290260 (44:7444)
  • [R] E. R. Reifenberg, Solution of the Plateau problem for m-dimensional surfaces of varying topological type, Acta Math. 104 (1960), 1-82. MR 0114145 (22:4972)
  • [T] Jean Taylor, The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces, Ann. of Math. (2) 103 (1976), 489-539. MR 0428181 (55:1208a)
  • [W1] Brian White, Regularity of the singular sets in immiscible fluid interfaces and in solutions to other plateau-type problems, Proc. Centre Math. Anal., Austral. Nat. Univ., Canberra, 1985, pp. 244-249. MR 857670
  • [W2] -, Regularity of singular sets for plateau-type problems (in preparation).

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 49Q05, 49Q20

Retrieve articles in all journals with MSC: 49Q05, 49Q20


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1994-1231039-1
Keywords: Area-minimizing, length of singular curves
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society