Surfaces minimizing area plus length of singular curves

Morgan, Frank

doi:10.1090/S0002-9939-1994-1231039-1

Surfaces minimizing area plus length of singular curves
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Proc. Amer. Math. Soc. 122 (1994), 1153-1161 Request permission

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

Jeff Abrahamson, Curves length-minimizing modulo $v$ in $\textbf {R}^n$, Michigan Math. J. 35 (1988), no. 2, 285–290. MR 959275, DOI 10.1307/mmj/1029003755
William K. Allard, On boundary regularity for Plateau’s problem, Bull. Amer. Math. Soc. 75 (1969), 522–523. MR 241802, DOI 10.1090/S0002-9904-1969-12229-9
William K. Allard, On the first variation of a varifold: boundary behavior, Ann. of Math. (2) 101 (1975), 418–446. MR 397520, DOI 10.2307/1970934
F. J. Almgren Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. Amer. Math. Soc. 4 (1976), no. 165, viii+199. MR 420406, DOI 10.1090/memo/0165
Hans Wilhelm Alt, Die Existenz einer Minimalfläche mit freiem Rand vorgeschriebener Länge, Arch. Rational Mech. Anal. 51 (1973), 304–320 (German). MR 336528, DOI 10.1007/BF00250537

The search for boundary-minimizing polytopes in two and three dimensions

Kenneth A. Brakke, Minimal cones on hypercubes, J. Geom. Anal. 1 (1991), no. 4, 329–338. MR 1129346, DOI 10.1007/BF02921309
Kenneth A. Brakke, The surface evolver, Experiment. Math. 1 (1992), no. 2, 141–165. MR 1203871

Networks minimizing length plus the number of Steiner points

Edith A. Cook, Free boundary regularity for surfaces minimizing $\textrm {Area}(S)+c\,\textrm {Area}(\partial S)$, Trans. Amer. Math. Soc. 290 (1985), no. 2, 503–526. MR 792809, DOI 10.1090/S0002-9947-1985-0792809-6

Su alcuni problemi comuni all’analisi e alla geometria

IX-Suppl.

Klaus Ecker, Area-minimizing integral currents with movable boundary parts of prescribed mass, Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (1989), no. 4, 261–293 (English, with French summary). MR 998604
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
Herbert Federer, 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 260981, DOI 10.1090/S0002-9904-1970-12542-3
L. Fejes Tóth, Regular figures, A Pergamon Press Book, The Macmillan Company, New York, 1964. MR 0165423
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
Stefan Hildebrandt, Ein einfacher Beweis für die Regularität der Lösungen gewisser zweidimensionaler Variationsprobleme unter freien Randbedingungen, Math. Ann. 194 (1971), 316–331 (German). MR 291927, DOI 10.1007/BF01350132
S. Hildebrandt, On two isoperimetric problems with free boundary conditions, Variational methods for free surface interfaces (Menlo Park, Calif., 1985) Springer, New York, 1987, pp. 43–51. MR 872887
Gary Lawlor and Frank Morgan, Paired calibrations applied to soap films, immiscible fluids, and surfaces or networks minimizing other norms, Pacific J. Math. 166 (1994), no. 1, 55–83. MR 1306034
Z. A. Melzak, Problems connected with convexity, Canad. Math. Bull. 8 (1965), 565–573. MR 196594, DOI 10.4153/CMB-1965-040-x
Frank Morgan, Clusters minimizing area plus length of singular curves, Math. Ann. 299 (1994), no. 4, 697–714. MR 1286892, DOI 10.1007/BF01459806
Frank Morgan, Geometric measure theory, Academic Press, Inc., Boston, MA, 1988. A beginner’s guide. MR 933756

Destabilization of the tetrahedral point junction by positive triple junction line energy

25

David Mumford and Jayant Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math. 42 (1989), no. 5, 577–685. MR 997568, DOI 10.1002/cpa.3160420503
Johannes C. C. Nitsche, The regularity of minimal surfaces on the movable parts of their boundaries, Indiana Univ. Math. J. 21 (1971/72), 505–513. MR 290260, DOI 10.1512/iumj.1971.21.21040
E. R. Reifenberg, Solution of the Plateau Problem for $m$-dimensional surfaces of varying topological type, Acta Math. 104 (1960), 1–92. MR 114145, DOI 10.1007/BF02547186
Jean E. Taylor, The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces, Ann. of Math. (2) 103 (1976), no. 3, 489–539. MR 428181, DOI 10.2307/1970949
Brian White, Regularity of the singular sets in immiscible fluid interfaces and solutions to other Plateau-type problems, Miniconference on geometry and partial differential equations (Canberra, 1985) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 10, Austral. Nat. Univ., Canberra, 1986, pp. 244–249. MR 857670

Regularity of singular sets for plateau-type problems