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Sets of Finite Perimeter and Geometric Variational Problems
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Book description

The marriage of analytic power to geometric intuition drives many of today's mathematical advances, yet books that build the connection from an elementary level remain scarce. This engaging introduction to geometric measure theory bridges analysis and geometry, taking readers from basic theory to some of the most celebrated results in modern analysis. The theory of sets of finite perimeter provides a simple and effective framework. Topics covered include existence, regularity, analysis of singularities, characterization and symmetry results for minimizers in geometric variational problems, starting from the basics about Hausdorff measures in Euclidean spaces and ending with complete proofs of the regularity of area-minimizing hypersurfaces up to singular sets of codimension 8. Explanatory pictures, detailed proofs, exercises and remarks providing heuristic motivation and summarizing difficult arguments make this graduate-level textbook suitable for self-study and also a useful reference for researchers. Readers require only undergraduate analysis and basic measure theory.

Reviews

'The book is a clear exposition of the theory of sets of finite perimeter, that introduces this topic in a very elegant and original way, and shows some deep and important results and applications … Although most of the results contained in this book are classical, some of them appear in this volume for the first time in book form, and even the more classical topics which one may find in several other books are presented here with a strong touch of originality which makes this book pretty unique … I strongly recommend this excellent book to every researcher or graduate student in the field of calculus of variations and geometric measure theory.'

Alessio Figalli Source: Canadian Mathematical Society Notes

'The first aim of the book is to provide an introduction for beginners to the theory of sets of finite perimeter, presenting results concerning the existence, symmetry, regularity and structure of singularities in some variational problems involving length and area … The secondary aim … is to provide a multi-leveled introduction to the study of other variational problems … an interested reader is able to enter with relative ease several parts of geometric measure theory and to apply some tools from this theory in the study of other problems from mathematics … This is a well-written book by a specialist in the field … It provides generous guidance to the reader [and] is recommended … not only to beginners who can find an up-to-date source in the field but also to specialists … It is an invitation to understand and to approach some deep and difficult problems from mathematics and physics.'

Vasile Oproiu Source: Zentralblatt MATH

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Contents


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References
References
[AB90] L., Ambrosio and A., Braides. Functionals defined on partitions in sets of finite perimeter II: Semicontinuity, relaxation and homogenization. J. Math. P ures Appl., 69:307–333, 1990.
[AC09] F., Alter and V., Caselles. Uniqueness of the Cheeger set of a convex body. Nonlinear Anal., 70(1):32–44, 2009.
[ACC05] F., Alter, V., Caselles, and A., Chambolle. A characterization of convex calibrable sets in ℝn. Math. Ann., 332(2):329–366, 2005.
[ACMM01] L., Ambrosio, V., Caselles, S., Masnou, and J. M., Morel. Connected components of sets of finite perimeter and applications to image processing. J. Eur. Math. Soc. (JEMS), 3(1):39–92, 2001.
[AFP00] L., Ambrosio, N., Fusco, and D., Pallara. Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. New York, The Clarendon Press, Oxford University Press, 2000. xviii + 434 pp.
[All72] W. K., Allard. On the first variation of a varifold. Ann. Math., 95:417–491, 1972.
[All75] W. K., Allard. On the frst variation of a varifold: boundary behaviour. Ann.Math., 101:418–446, 1975.
[Alm66] F. J., Almgren Jr. Plateau's Problem. An Invitation to Varifold Geometry. Mathematics Monograph Series. New York, Amsterdam, W. A. Benjamin, Inc., 1966. xii + 74 pp.
[Alm68] F. J., Almgren Jr, Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure. Ann. Math., 87:321–391, 1968.
[Alm76] F. J., Almgren Jr. Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints. Mem. Amer. Math. Soc., 4(165): viii + 199 pp, 1976.
[Alm00] F. J., Almgren Jr, Almgren's Big Regularity Paper. Q-valued Functions Minimizing Dirichlet's Integral and the Regularity of Area-minimizing Rectifable Currents up to Codimension 2, Volume 1 of World Scientific Monograph Series in Mathematics. River Edge, NJ, World Scientifc Publishing Co., Inc., 2000. xvi + 955 pp.
[AM98] L., Ambrosio and C., Mantegazza. Curvature and distance function from a manifold. J. Geom. Anal., 8:723–748, 1998.
[Amb97] L., Ambrosio. Corso Introduttivo alla Teoria Geometrica della Misura ed alle Superfci Minime. Pisa, Edizioni Scuola Normale Superiore di Pisa, 1997.
[AP99] L., Ambrosio and E., Paolini. Partial regularity for quasi minimizers of perimeter. Ricerche Mat., 48:167–186, 1999.
[AP01] L., Ambrosio and E., Paolini. Errata-corrige: “Partial regularity for quasi minimizers of perimeter”. Ricerche Mat., 50:191–193, 2001.
[BCCN06] G., Bellettini, V., Caselles, A., Chambolle, and M., Novaga. Crystalline mean curvature fow of convex sets. Arch. Rational Mech. Anal., 179(1): 109–152, 2006.
[BdC84] J. L., Barbosa and M. do, Carmo. Stability of hypersurfaces with constant mean curvature. Math. Z., 185(3):339–353, 1984.
[BDGG69] E., Bombieri, E., De Giorgi, and E., Giusti. Minimal cones and the Bernstein problem. Invent. Math., 7:243–268, 1969.
[BGM03] E., Barozzi, E., Gonzalez, and U., Massari. The mean curvature of a Lipschitz continuous manifold. Rend. Mat. Acc. Lincei s. 9, 14:257–277, 2003.
[BGT87] E., Barozzi, E., Gonzalez, and I., Tamanini. The mean curvature of a set of finite perimeter. Proc. Amer. Math. Soc., 99(21):313–316, 1987.
[Bin78] D., Bindschadler. Absolutely area minimizing singular cones of arbitrary codimension. Trans. Amer. Math. Soc., 243:223–233, 1978.
[BM94] J. E., Brothers and F., Morgan. The isoperimetric theorem for general integrands. Mich. Math. J., 41(31):419–431, 1994.
[BNP01a] G., Bellettini, M., Novaga, and M., Paolini. On a crystalline variational problem I: First variation and global L∞ regularity. Arch. Ration. Mech. Anal., 157(3):165–191, 2001.
[BNP01b] G., Bellettini, M., Novaga, and M., Paolini. On a crystalline variational problem II: BV regularity and structure of minimizers on facets. Arch. Ration. Mech. Anal., 157(3):193–217, 2001.
[Bom82] E., Bombieri. Regularity theory for almost minimal currents. Arch. Ration. Mech. Anal., 7(7):99–130, 1982.
[Bon09] M., Bonacini. Struttura delle m-bolle di Perimetro Minimo e il Teorema della Doppia Bolla Standard. Master's Degree thesis, Universit àdegli Studi di Modena e Reggio Emilia, 2009.
[BNR03] M., Bellettini, M., Novaga, and G., Riey. First variation of anisotropic energies and crystalline mean curvature for partitions. Interfaces Free Bound., 5(3):331–356, 2003.
[CC06] V., Caselles and A., Chambolle. Anisotropic curvature-driven flow of convex sets. Nonlinear Anal., 65(8):1547–1577, 2006.
[CCF05] M., Chlebik, A., Cianchi, and N., Fusco. The perimeter inequality under Steiner symmetrization: cases of equality. Ann. Math., 162:525–555, 2005.
[CCN07] V., Caselles, A., Chambolle, and M., Novaga. Uniqueness of the Cheeger set of a convex body. Pacifc J. Math., 232(1):77–90, 2007.
[CCN10] V., Caselles, A., Chambolle, and M., Novaga. Some remarks on uniqueness and regularity of Cheeger sets. Rend. Semin. Mat. Univ. Padova, 123: 191201, 2010.
[CF10] S. J., Cox and E., Flikkema. The minimal perimeter for n confned de formable bubbles of equal area. Electron. J. Combin., 17(1), 2010.
[Dav04] A., Davini. On calibrations for Lawson's cones. Rend. Sem. Mat. Univ. Padova, 111:55–70, 2004.
[Dav09] G., David. Hölder regularity of two-dimensional almost-minimal sets in ℝn. Ann. Fac. Sci. Toulouse Math.(6), 18(1):65–246, 2009.
[Dav10] G., David. C1+α-regularity for two-dimensional almost-minimal sets in ℝn. J. Geom. Anal., 20(4):837–954, 2010.
[DG54] E., De Giorgi. Su una teoria generale della misura (r - 1)-dimensionale in uno spazio ad r-dimensioni. Ann. Mat. Pura Appl. (4), 36:191–213, 1954.
[DG55] E., De Giorgi. Nuovi teoremi relativi alle misure (r - 1)-dimensionali in uno spazio ad r-dimensioni. Ricerche Mat., 4:95–113, 1955.
[DG58] E., De Giorgi. Sulla proprietà isoperimetrica dell'ipersfera, nella classe degli insiemi aventi frontiera orientata di misura finita. Atti Accad. Naz. Lincei. Mem. Cl. Sci. Fis. Mat. Nat. Sez.I(8), 5:33–44, 1958.
[DG60] E., De Giorgi. Frontiere Orientate di Misura Minima. Seminario di Matematica della Scuola Normale Superiore di Pisa. Editrice Tecnico Scientifca, Pisa, 1960. 57 pp.
[Din44] A., Dinghas. Über einen geometrischen Satz von Wulf für die Gle-ichgewichtsform von Kristallen. Z. Kristallogr. Mineral. Petrogr. Abt. A., 105:304–314, 1944.
[DL08] C., De Lellis. Rectifable Sets, Densities and Tangent Measures. Z ürich Lectures in Advanced Mathematics. Zürich, European Mathematical Society (EMS), 2008. vi + 127 pp.
[DLS11a] C., De Lellis and E. N., Spadaro. Center manifold: a study case. Disc. Cont. Din. Syst. A, 31(4):1249–1272, 2011.
[DLS11b] C., De Lellis and E. N., Spadaro. q-valued functions revisited. Mem. Amer. Math. Soc., 211: vi + 79 pp., 2011.
[DM95] G., Dolzmann and S., Müller. Microstructures with finite surface energy: the two-well problem. Arch. Rational Mech. Anal., 132(2):101–141, 1995.
[DPM12] G., De Philippis and F., Maggi. Sharp stability inequalities for the Plateau problem. Preprint cvgmt.sns.it/paper/1715/, 2012.
[DPP09] G., De Philippis and E., Paolini. A short proof of the minimality of Simons cone. Rend. Sem. Mat. Univ. Padova, 121:233–241, 2009.
[DS93] F., Duzaar and K., Stefen. Boundary regularity for minimizing currents with prescribed mean curvature. Calc. Var., 1:355–406, 1993.
[DS02] F., Duzaar and K., Stefen. Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals. J. Reine Angew. Math., 564:73–138, 2002.
[EG92] L. C., Evans and R. F., Gariepy. Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. Boca Raton, FL, CRC Press, 1992. viii + 268 pp.
[Eva98] L. C., Evans. Partial Differential Equations, volume 19 of Graduate Studies in Mathematics. Providence, RI, American Mathematical Society, 1998. xviii + 662 pp.
[FAB+93] J., Foisy, M., Alfaro, J., Brock, N., Hodges, and J., Zimba. The standard double soap bubble in ℝ2 uniquely minimizes perimeter. Pacific J. Math., 159(1), 1993.
[Fal86] K. J., Falconer. The Geometry of Fractal Sets, volume 85 of CambridgeTracts in Mathematics. Cambridge, Cambridge University Press, 1986.xiv+162 pp.
[Fal90] K. J., Falconer. Fractal Geometry. Mathematical Foundations and Applications. Chichester, John Wiley and Sons, Ltd., 1990. xxii+288 pp.
[Fal97] K. J., Falconer. Techniques in Fractal Geometry. Chichester, John Wiley and Sons, Ltd., 1997. xviii+256 pp.
[Fed69] H., Federer. Geometric Measure Theory, volume 153 of Die Grundlehren der mathematischen Wissenschaften. New York, Springer-Verlag New York Inc., 1969. xiv+676 pp.
[Fed70] H., Federer. The singular sets of area minimizing rectifable currents with codimension one and of area minimizing fat chains modulo two with arbitrary codimension. Bull. Amer. Math. Soc., 76:767–771, 1970.
[Fed75] H., Federer. Real flat chains, cochains and variational problems. Indiana Univ. Math. J., 24:351–407, 1974/75.
[FF60] H., Federer and W. H., Fleming. Normal and intregral currents. Ann. Math., 72:458–520, 1960.
[Fin86] R., Finn. Equilibrium Capillary Surfaces, volume 284 of Die Grundlehren der mathematischen Wissenschaften. New York, Springer-Verlag New York Inc., 1986. xi+245 pp.
[Fle62] W. H., Fleming. On the oriented plateau problem. Rend. Circ. Mat. Palermo (2), 11:69–90, 1962.
[FM91] I., Fonseca and S., Müller. A uniqueness proof for the Wulff theorem. Proc. R. Soc. Edinb. Sect. A, 119(12):125–136, 1991.
[FM92] I., Fonseca and S., Müller. Quasi-convex integrands and lower semicontinuity in L1. SIAM J. Math. Anal., 23(5):1081–1098, 1992.
[FM11] A., Figalli and F., Maggi. On the shape of liquid drops and crystals in the small mass regime. Arch. Rat. Mech. Anal., 201:143–207, 2011.
[FMP08] N., Fusco, F., Maggi and A., Pratelli. The sharp quantitative isoperimetric inequality. Ann. Math., 168:941–980, 2008.
[FMP10] A, Figalli, F., Maggi and A., Pratelli. A mass transportation approach to quantitative isoperimetric inequalities. Inv. Math., 182(1):167–211, 2010.
[Fus04] N., Fusco. The classical isoperimetric theorem. Rend. Accad. Sci. Fis. Mat. Napoli (4), 71:63–107, 2004.
[Gar02] R. J., Gardner. The Brunn–Minkowski inequality. Bull. Am. Math. Soc. (NS), 39(3):355–405, 2002.
[Giu80] E., Giusti. The pendent water drop. A direct approach. Bollettino UMI, 17-A:458–465, 1980.
[Giu81] E., Giusti. The equilibrium confguration of liquid drops. J. Reine Angew. Math., 321:53–63, 1981.
[Giu84] E., Giusti. Minimal Surfaces and Functions of Bounded Variation,volume 80 of Monographs in Mathematics. Basel, Birkhäuser Verlag, 1984. xii+240 pp.
[GM94] E. H. A., Gonzalez and U., Massari. Variational mean curvatures. Rend. Sem. Mat. Univ. Politec. Torino, 52(1):128, 1994.
[GM05] M., Giaquinta and L., Martinazzi. An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs, volume 2 of Lecture Notes. Scuola Normale Superiore di Pisa (New Series). Pisa, Edizioni della Normale, 2005. xiv+676 pp.
[GMS98a] M., Giaquinta, G., Modica, and J., Soucek. Cartesian Currents in the Calculus of Variations. I. Cartesian Currents, volume 37 of Ergebnisse derMathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics. Berlin, Springer-Verlag, 1998. xxiv+711 pp.
[GMS98b] M., Giaquinta, G., Modica, and J., Soucek. Cartesian Currents in the Calculus of Variations. II. Variational Integrals, volume 38 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics. Berlin, Springer-Verlag, 1998. xxiv+697 pp.
[GMT80] E., Gonzalez, U., Massari, and I., Tamanini. Existence and regularity for the problem of a pendent liquid drop. Pacific Journal of Mathematics, 88(2):399–420, 1980.
[GMT93] E. H. A., Gonzales, U., Massari, and I., Tamanini. Boundaries of prescribed mean curvature. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 4(3):197–206, 1993.
[Gon76] E., Gonzalez. Sul problema della goccia appoggiata. Rend. Sem. Mat. Univ. Padova, 55:289–302, 1976.
[Gon77] E., Gonzalez. Regolarità per il problema della goccia appoggiata. Rend. Sem. Mat. Univ. Padova, 58:25–33, 1977.
[Grü87] M., Grüter. Boundary regularity for solutions of a partitioning problem. Arch. Rat. Mech. Anal., 97:261–270, 1987.
[GT77] E., Gonzalez and I., Tamanini. Convessità della goccia appoggiata. Rend. Sem. Mat. Univ. Padova, 58:35–43, 1977.
[GT98] D., Gilbarg and N. S., Trudinger. Elliptic Partial Differential E quations of Second Order. Berlin; New York, Springer, 1998. xiii+517 pp.
[Hal01] T. C., Hales. The honeycomb conjecture. Discrete Comput. Geom., 25(1):1–22, 2001.
[Har77] R., Hardt. On boundary regularity for integral currents or flat chains modulo two minimizing the integral of an elliptic integrand. Comm. Part. Diff. Equ., 2:1163–1232, 1977.
[HL86] R., Hardt and F. H., Lin. Tangential regularity near the C1 boundary. Proc. Symp. Pure Math., 44:245–253, 1986.
[HMRR02] M., Hutchings, F., Morgan, M., Ritoré, and A., Ros. Proof of the double bubble conjecture. Ann. of Math. (2), 155(2):459–489, 2002.
[HS79] R., Hardt and L., Simon. Boundary regularity and embedded solutions for the oriented Plateau problem. Ann. Math., 110:439–486, 1979.
[Hut81] J. E., Hutchinson. Fractals and self-similarity. Indiana Univ. M ath. J., 30(5):713–747, 1981.
[Hut97] M., Hutchings. The structure of area-minimizing double bubbles. J. Geom. Anal., 7:285–304, 1997.
[Kno57] H., Knothe. Contributions to the theory of convex bodies. Mich. M ath. J., 4:39–52, 1957.
[KP08] S. G., Krantz and H. R., Parks. Geometric Integration Theory, volume 80 of Cornerstones. Boston, MA, Birkhäuser Boston, Inc., 2008. xvi+339 pp.
[KS00] D., Kinderlehrer and G., Stampacchia. An Introduction to Variational Inequalities and their Applications, volume 31 of Classics in Applied Mathematics. Philadelphia, PA, Society for Industrial and Applied Mathematics (SIAM), 2000.
[Law72] H. B., Lawson Jr, The equivariant Plateau problem and interior regularity. Trans. Amer. Math. Soc., 173:231–249, 1972.
[Law88] G. R., Lawlor. A Sufficient Criterion for a Cone to be Area Minimizing. PhD thesis, Stanford University, 1988.
[Leo00] G. P., Leonardi. Blow-up of oriented boundaries. Rend. Sem. Mat. Univ. Padova, 103:211–232, 2000.
[Leo01] G. P., Leonardi. Infiltrations in immiscible fulids systems. Proc. Roy. Soc. Edinburgh Sect. A, 131(2):425–436, 2001.
[Mas74] U., Massari. Esistenza e regolarità delle ipersuperfice di curvatura media assegnata in ℝn. Arch. Rational Mech. Anal., 55:357–382, 1974.
[Mas75] U., Massari. Frontiere orientate di curvatura media assegnata in Lp. Rend. Sem. Mat. Univ. Padova, 53:37–52, 1975.
[Mat95] P., Mattila. Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifability, volume 44 of Cambridge Studies in Advanced Mathematics. Cambridge, Cambridge University Press, 1995. xii+343 pp.
[McC98] R. J., McCann. Equilibrium shapes for planar crystals in an external field. Comm. Math. Phys., 195(3):699–723, 1998.
[Mir65] M., Miranda. Sul minimo dell'integrale del gradiente di una funzione. Ann. Scuola Norm. Sup. Pisa (3), 19:626–665, 1965.
[Mir67] M., Miranda. Comportamento delle successioni convergenti di frontiere minimali. Rend. Sem. Mat. Univ. Padova, 38:238257, 1967.
[Mir71] M., Miranda. Frontiere minimali con ostacoli. Ann. Univ. Ferrara Sez. VII (N.S.), 16:29–37, 1971.
[MM83] U., Massari and M., Miranda. A remark on minimal cones. Boll. Un. Mat. Ital. A (6), 2(1):123–, 125.
[MM84] U., Massari and M., Miranda. Minimal Surfaces of Codimension One. North-Holland Mathematics Studies, 91. Notas de Matematica [Mathematical Notes], 95. Amsterdam, North-Holland Publishing Co., 1984.xiii+243 pp.
[Mor66] C. B., Morrey Jr, Multiple Integrals in the Calculus of Variations. Berlin, Heidelberg, New York, Springer-Verlag, 1966.
[Mor90] F., Morgan. A sharp counterexample on the regularity of Φ-minimizing hypersurfaces. Bull. Amer. Math. Soc. (N.S.), 22(2):295–299. 1990
[Mor94] F., Morgan. Soap bubbles in ℝ2 and in surfaces. Pacific J. Math., 165(2):347–361, 1994.
[Mor09] F., Morgan. Geometric Measure Theory. A Beginner's Guide. Fourth edition. Amsterdam, Elsevier/Academic Press, 2009. viii+249 pp.
[MS86] V. D., Milman and G., Schechtman. Asymptotic Theory of Finitedimensional Normed Spaces. With an appendix by M. Gromov. Number 1200 in Lecture Notes in Mathematics. Berlin, Springer-Verlag, 1986. viii+156 pp.
[MW02] F., Morgan and W., Wichiramala. The standard double bubble is the unique stable double bubble in ℝ2. Proc. Amer. Math. Soc., 130(9):2745–2751, 2002.
[Pao98] E., Paolini. Regularity for minimal boundaries in ℝnwith mean curvature in ℝn. Manuscripta Math., 97(1):15–35.1998.
[Rei60] E. R., Reifenberg. Solution of the Plateau problem for m-dimensional surfaces of varying topological type. Acta Math., 104:1–92, 1960.
[Rei64a] E. R., Reifenberg. An epiperimetric inequality related to the analyticity of minimal surfaces. Ann. of Math., 80(2):1–14, 1964.
[Rei64b] E. R., Reifenberg. On the analyticity of minimal surfaces. Ann. of Math., 80(2):15–21, 1964.
[Rei08] B. W., Reichardt. Proof of the double bubble conjecture in ℝn. J. Geom. Anal., 18(1):172–191, 2008.
[RHLS03] B. W., Reichardt, C., Heilmann, Y. Y., Lai, and A., Spielman. Proof of the double bubble conjecture in ℝ4 and certain higher dimensional cases. Pacific J. Math., 208(2), 2003.
[Roc70] R. T., Rockafellar. Convex Analysis, volume 28 of Princeton Mathematical Series. Princeton, NJ, Princeton University Press, 1970. xviii+451 pp.
[Sim68] J., Simons. Minimal varieties in Riemannian manifolds. Ann. Math., 88(1):62–105, 1968.
[Sim73] P., Simoes. A Class of Minimal Cones in ℝn, n ≥ 8, that Minimize Area. PhD thesis, University of California, Berkeley, 1973.
[Sim83] L., Simon. Lectures on Geometric Measure Theory, volume 3 of Proceedings of the Centre for Mathematical Analysis. Canberra, Australian National University, Centre for Mathematical Analysis, 1983. vii+272 pp.
[Sim96] L., Simon. Theorems on Regularity and Singularity of Energy Minimizing Maps. Basel, Boston, Berlin, Birkhaüser-Verlag, 1996.
[Spa09] E. N., Spadaro. Q-valued Functions and Approximation of Minimal Currents. PhD thesis, Universität Zürich, 2009.
[Spe11] D., Spector. Simple proofs of some results of Reshetnyak. Proc. A mer. Math. Soc., 139:1681–1690, 2011.
[Spi65] M., Spivak. Calculus on Manifolds. A Modern Approach to Classical Theorems of Advanced Calculus. NewYork, Amsterdam, W.A.Benjamin, Inc., 1965. xii+144 pp.
[SS82] R., Schoen and L., Simon. A new proof of the regularity theorem for rectifiable currents which minimize parametric elliptic functionals. Indiana Univ. Math. J., 31(3):415–434, 1982.
[SSA77] R., Schoen, L., Simon, and F. J., Almgren JrRegularity and singularity estimates on hypersurfaces minimizing parametric elliptic variational in tegrals. i, ii. Acta Math., 139(3–4):217–265, 1977.
[Str79] K., Stromberg. The Banach–Tarski paradox. Amer. Math. Monthly, 86(3):151–161, 1979.
[SZ98] P., Sternberg and K., Zumbrun. A Poincaré inequality with applications to volume-constrained area-minimizing surfaces. J. Reine Angew. Math., 503:63–85, 1998.
[SZ99] P., Sternberg and K., Zumbrun. On the connectedness of boundaries of sets minimizing perimeter subject to a volume constraint. Comm. Anal. Geom., 7(1):199–220, 1999.
[Tam82] I., Tamanini. Boundaries of Caccioppoli sets with Hölder-continuous normal vector. J. Reine Angew. Math., 334:27–39, 1982.
[Tam84] I., Tamanini. Regularity Results for Almost Minimal Oriented Hypersurfaces in ℝN. Quaderni del Dipartimento di Matematica dell'Università di Lecce. Lecce, Università di Lecce, 1984.
[Tam98] I., Tamanini. Counting elements of least-area partitions. Atti Sem. Mat. Fis. Univ. Modena, 46(suppl.):963–969, 1998.
[Tay76] J. E., Taylor. The structure of singularities in soap-bubble-like and soapflm-like minimal surfaces. Ann. of Math. (2), 103(3):489–539, 1976.
[Tay78] J. E., Taylor. Crystalline variational problems. Bull. Am. Math. Soc., 84(4):568–588, 1978.
[Wen91] H. C., Wente. A note on the stability theorem of J. L. Barbosa and M. do Carmo for closed surfaces of constant mean curvature. Pacific J. Math., 147(2):375–379, 1991.
[Whi] B., White. Regularity of singular sets for Plateau-type problems. See the announcement on author's webpage, math.stanford.edu/∼white/bibrev.htm.
[Whi96] B., White. Existence of least-energy confgurations of immiscible fluids. J. Geom. Analysis, 6:151–161, 1996.
[Wic04] W., Wichiramala. Proof of the planar triple bubble conjecture. J. Reine Angew. Math., 567:1–49, 2004.
[WP94] D., Weaire and R., Phelan. A counter-example to Kelvin's conjecture on minimal surfaces. Phil. Mag. Lett., 69:107–110, 1994.
[Wul01] G., Wulf. Zur Frage der Geschwindigkeit des Wachsturms und der aufösungder Kristall-Flächen. Z. Kristallogr., 34:449–530, 1901.

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