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Colloquium lectures on geometric measure theory. (English) Zbl 0392.49021


MSC:

49Q15 Geometric measure and integration theory, integral and normal currents in optimization
49Q20 Variational problems in a geometric measure-theoretic setting
26A45 Functions of bounded variation, generalizations
28A75 Length, area, volume, other geometric measure theory
49Q99 Manifolds and measure-geometric topics
58A15 Exterior differential systems (Cartan theory)
49Q05 Minimal surfaces and optimization
53C65 Integral geometry
49-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control

Citations:

Zbl 0176.008
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References:

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[33] Ronald Gariepy, Geöcze area and a convergence property, Proc. Amer. Math. Soc. 34 (1972), 469 – 474. · Zbl 0259.28015
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[36] Casper Goffman, An example in surface area, J. Math. Mech. 19 (1969/1970), 321 – 326. · Zbl 0197.03701
[37] Casper Goffman and William P. Ziemer, Higher dimensional mappings for which the area formula holds, Ann. of Math. (2) 92 (1970), 482 – 488. · Zbl 0204.08001
[38] Robert M. Hardt, Slicing and intersection theory for chains associated with real analytic varieties, Acta Math. 129 (1972), 75 – 136. · Zbl 0234.32005
[39] Robert M. Hardt, Slicing and intersection theory for chains modulo \? associated with real analytic varieties, Trans. Amer. Math. Soc. 183 (1973), 327 – 340. · Zbl 0267.28010
[40] Robert M. Hardt, Homology theory for real analytic and semianalytic sets, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), no. 1, 107 – 148. · Zbl 0309.32004
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[42] Robert M. Hardt, Uniqueness of nonparametric area minimizing currents, Indiana Univ. Math. J. 26 (1977), no. 1, 65 – 71. · Zbl 0333.49043
[43] Robert M. Hardt, On boundary regularity for integral currents or flat chains modulo two minimizing the integral of an elliptic integrand, Comm. Partial Differential Equations 2 (1977), no. 12, 1163 – 1232. · Zbl 0385.49025
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[46] F. Reese Harvey and H. Blaine Lawson Jr., On boundaries of complex analytic varieties. I, Ann. of Math. (2) 102 (1975), no. 2, 223 – 290. · Zbl 0317.32017
[47] Reese Harvey and Bernard Shiffman, A characterization of holomorphic chains, Ann. of Math. (2) 99 (1974), 553 – 587. · Zbl 0287.32008
[48] S. Kar, The \((\varphi , 1)\) rectifiable subsets of Euclidean space, Indiana Ph.D. Thesis, 1975. · Zbl 0376.28021
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[51] Robert V. Kohn, An example concerning approximate differentiation, Indiana Univ. Math. J. 26 (1977), no. 2, 393 – 397. · Zbl 0349.26007
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[53] H. Blaine Lawson Jr., The stable homology of a flat torus, Math. Scand. 36 (1975), 49 – 73. Collection of articles dedicated to Werner Fenchel on his 70th birthday. · Zbl 0346.53040
[54] H. B. Lawson, Jr. and R. Osserman, Nonexistence, nonuniqueness and irregularity of solutions to the minimal surface system, Acta Math, (to appear).
[55] H. Blaine Lawson Jr. and James Simons, On stable currents and their application to global problems in real and complex geometry, Ann. of Math. (2) 98 (1973), 427 – 450. · Zbl 0283.53049
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[58] Norman G. Meyers and William P. Ziemer, Integral inequalities of Poincaré and Wirtinger type for BV functions, Amer. J. Math. 99 (1977), no. 6, 1345 – 1360. · Zbl 0416.46025
[59] M. Miranda, Un principio di massimo forte per le frontiere minimali e una sua applicazione alla risoluzione del problema al contorno per l’equazione delle superfici di area minima, Rend. Sem. Mat. Univ. Padova 45 (1971), 355 – 366 (Italian). · Zbl 0266.49034
[60] Frank Morgan, A smooth curve in \?\(^{4}\) bounding a continuum of area minimizing surfaces, Duke Math. J. 43 (1976), no. 4, 867 – 870. · Zbl 0338.53003
[61] F. Morgan, Almost every smooth curve in R3 bounds a unique area minimizing surface, Princeton Ph.D. Thesis, 1977.
[62] Johannes C. C. Nitsche, Vorlesungen über Minimalflächen, Springer-Verlag, Berlin-New York, 1975 (German). Die Grundlehren der mathematischen Wissenschaften, Band 199. · Zbl 0319.53003
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[65] Harold R. Parks, A method for computing non-parametric area minimizing surfaces over \?-dimensional domains, together with a priori error estimates, Indiana Univ. Math. J. 26 (1977), no. 4, 625 – 643. · Zbl 0362.49030
[66] Harold Parks, Explicit determination of area minimizing hypersurfacess, Duke Math. J. 44 (1977), no. 3, 519 – 534. · Zbl 0385.49026
[67] Sandra O. Paur, Stokes’ theorem for integral currents modulo \?, Amer. J. Math. 99 (1977), no. 2, 379 – 388. · Zbl 0385.49024
[68] S. O. Paur, An estimate of the density at the boundary of an integral current modulo v · Zbl 0368.53047
[69] Jon T. Pitts, Existence and regularity of minimal surfaces on Riemannian manifolds, Bull. Amer. Math. Soc. 82 (1976), no. 3, 503 – 504. · Zbl 0329.49029
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[76] Jean E. Taylor, Regularity of the singular sets of two-dimensional area-minimizing flat chains modulo 3 in \?³, Invent. Math. 22 (1973), 119 – 159. · Zbl 0278.49046
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