Geometry V pp 239-266

# The Minimal Surface Equation

• Leon Simon
Chapter
Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 90)

## Abstract

The minimal surface equation (MSE) for functions u: Ω → ℝ, Ω a domain of ℝ2, can be written
$$\left( {1 + u_{}^2} \right){u_{xx}} - 2{u_x}{u_y}{u_{xy}} + \left( {1 + u_x^2} \right){u_{yy}} = 0$$
or equivalently $${u_{xx}} + {u_{yy}} - {\left( {1 + |Du{|^2}} \right)^{ - 1}}\left( {u_x^2{u_{xx}} + 2{u_x}{u_y}{u_{xy}} + u_y^2{u_{yy}}} \right) = 0$$ where $${u_x} = \frac{{\partial u\left( {x,y} \right)}}{{\partial x}},{u_y} = \frac{{\partial u\left( {x,y} \right)}}{{\partial y}}$$. Generally, for domains Ω ⊂ ℝ n and functions Ω → ℝ depending on the n variables (x 1, …, x n ) ∈ Ω, n ≥ 2, the MSE can be written
$$\sum\limits_{i,j = 1}^n {\left( {{\delta _{ij}} - \frac{{{u_i}{u_j}}}{{\left( {1 + |Du{|^2}} \right)}}} \right){u_{ij}} = 0}$$
where $${u_i} = {D_i}u \equiv \frac{{\partial u}}{{\partial {x^i}}}$$ and u ij = D i D j u. Notice that this is a quasilinear elliptic equation: that is, it is linear in the second derivatives, and the coefficient matrix $$\left( {{\delta _{ij}} - \frac{{{u_i}{u_j}}}{{\left( {1 + |Du{|^2}} \right)}}} \right)$$ is positive definite1 depending only on the derivatives up to first order. The equation can alternatively be written in “divergence form”
$$\sum\limits_{i = 1}^n {{D_i}} \left( {\frac{{{D_i}u}}{{\sqrt {1 + |Du{|^2}} }}} \right) = 0$$
(1)
which is readily checked using the chain rule and the fact that $$\frac{\partial }{{\partial {p_j}}}\left( {\frac{p}{{\sqrt {1 + |p{|^2}} }}} \right) = {\left( {1 + |p{|^2}} \right)^{ - 1/2}}\left( {{\delta _{ij}} - \frac{{{p_i}{p_j}}}{{1 + |p{|^2}}}} \right)$$.

## Keywords

Minimal Surface Dirichlet Problem Quasilinear Elliptic Equation Bernstein Theorem Minimal Surface Equation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. Almgren, F. (1966): Some interior regularity theorems for minimal surfaces and an extension of Bernstein’s theorem. Ann. Math. 84, 277–292 (1966), Zbl. 146, 119
2. Almgren, F., Schoen, R., and Simon, L. (1977): Regularity and Singularity Estimates for hypersurfaces minimizing parametric elliptic variational integrals. Acta Math. 139, 217–265 (1977), Zbl. 386.49030
3. Aronsson, G. (1968): On the partial differential equation (math). Ark. Mat. 7, 395–425 (1968), 162, 422
4. Bers, L. (1951): Isolated singularities of minimal surfaces. Ann. Math. 53, 364–386 (1951), Zbl. 43, 159
5. Bers, L. (1954): Non-linear elliptic equations without non-linear entire solutions. J. Rat. Mech. Anal. 3. 767–787 (1954), Zbl. 56, 321
6. Bernstein, S. (1910): Sur la généralisation du problème Dirichlet II. Math. Ann. 69, 82–136 (1910), Jbuch 41, 427
7. Bernstein, S. (1916): Über ein geometrisches Theorem und seine Anwendung auf die partiellen Differentialgleichungen vom ellipschen Typus. Math. Zeit. 26 (1927), 551–558 (translation of the original version in Comm. de la Soc. Math. de Kharkov 2-ème sér. 15 38–45 (1915–1917))
8. Bombieri, E., De Giorgi, E., and Miranda, M. (1969): Una maggiorazzione a priori relativa alle ipersuperfici minimali non parametriche. Arch. Rat. Mech. Anal. 32, 255–267 (1969). Zbl. 184, 328
9. Bombieri, E., De Giorgi, E., and Giusti, E. (1969): Minimal cones and the Bernstein problem. Invent. Math. 7, 243–268 (1969), Zbl. 183, 259
10. Bombieri, E., and Giusti, E. (1972): Harnack’s inequality for elliptic differential equations on minimal surfaces. Invent. Math. 15, 24–46 (1972), Zbl. 227.35021
11. Collin, P. (1990): Deux exemples de graphes de courbure moyenne constante sur une bande de ℝ2. C. R. Acad. Sci., Paris, Ser. I Math. 311, 539–542 (1990), Zbl. 716.53016
12. Collin, P., and Krust, R. (1991): Le problème de Dirichlet pour l’équation des surfaces minimales sur des domaines non bornés. Bull. Soc. Math. France 119, 443–462 (1991), Zbl. 754. 53013
13. Caffarelli, L., Nirenberg, L., and Spruck, J. (1990): On a form of Bernstein’s theorem. Analyse mathématique et applications 55–56, Gauthier-Villars, Paris 1990, Zbl. 668.35028Google Scholar
14. Courant, R., and Hilbert, D. (1962): Methods of Mathematical Physics, Vol. II. Interscience Publishers, New York 1962, Zbl. 99, 295
15. De Giorgi, E. (1957): Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 3, 25–43 (1957), Zbl. 84.319Google Scholar
16. De Giorgi, E. (1961): Frontiere orientate di misura minima. Sem. Mat. Sc. Norm. Sup. Pisa, 1–56 (1961)Google Scholar
17. De Giorgi, E. (1965): Una estensione del teorema di Bernstein. Ann. Sc. Norm. Sup. Pisa 19, 79–85 (1965), Zbl. 168, 98
18. De Giorgi, E., and Stampacchia, G. (1965): Sulle singolarità eliminabili delle ipersuperficie minimali. Atti Accad. Naz. Lincei, Rend., Cl. Sci. Fis. Mat. Nat. 38, 352–357 (1965), Zbl. 135, 400
19. Dierkes, U. (1993): A Bernstein result for energy minimizing hypersurfaces. Calc. Var. Partial Differ. Equ. 1, 37–54 (1993), Zbl. 819.35030
20. Dierkes, U., Hildebrandt, S., Küster, A., and Wohlrab, O. (1992): Minimal Surfaces, Vols. I, II. Springer-Verlag, Berlin Heidelberg New York 1992, Zbl. 777.53012, Zbl. 777.53013
21. Earp, R., and Rosenberg, H. (1989): The Dirichlet problem for the minimal surface equation on unbounded planar domains. J. Math. Pures Appl. 68, 163–183 (1989), Zbl. 696.49069
22. Ecker, K., and Huisken, G. (1989): Mean curvature evolution of entire graphs. Ann. Math. 130, 453–471 (1989), Zbl. 696.53036
23. Ecker, K., and Huisken, G. (1990): A Bernstein result for minimal graphs of controlled growth. J. Differ. Geom. 31, 397–400 (1990), Zbl. 696.53002
24. Ecker, K., and Huisken, G. (1991): Interior estimates for hypersurfaces moving by mean curvature. Invent. Math. 105, 547–569 (1991), Zbl. 725.53009
25. Federer, H. (1969): Geometric Measure Theory. Springer-Verlag, Berlin Heidelberg New York 1969, Zbl. 176, 8
26. Finn, R. (1953): A property of minimal surfaces. Proc. Nat. Acad. Sci. USA 39, 197–201 (1953), Zbl. 51, 125
27. Finn, R. (1954): On equations of minimal surface type. Ann. Math. 60, 397–416 (1954), Zbl. 58, 325
28. Finn, R. (1963): New estimates for equations of minimal surface type. Arch. Rat. Mech. Anal. 14, 337–375 (1963), Zbl. 133, 46
29. Finn, R. (1965): Remarks relevant to minimal surfaces and to surfaces of prescribed mean curvature. J. Anal. Math. 14, 139–160 (1965), Zbl. 163, 346
30. Finn, R. (1986): Equilibrium Capillary Surfaces. Springer-Verlag, Berlin Heidelberg New York 1986, Zbl. 583.35002
31. Finn, R., and Giusti, E. (1977): On nonparametric surfaces of constant mean curvature. Ann. Sc. Norm. Sup. Pisa 4, 13–31 (1977), Zbl. 343.53004
32. Finn, R., and Osserman, R. (1964): The gauss curvature of nonparametric minimal surfaces. J. Anal. Math. 12, 351–364 (1964), Zbl. 122, 164
33. Fischer-Colbrie, D. (1980): Some rigidity theorems for minimal submanifolds of the sphere. Acta. Math. 145, 29–46 (1980), Zbl. 464.53047
34. Fleming, W. (1962): On the oriented Plateau problem. Rend. Circ. Mat. Palermo 11, 69–90 (1962), Zbl. 107, 313
35. Gerhardt, C. (1974): Existence, regularity, and boundary behaviour of generalized surfaces of prescribed mean curvature. Math. Z. 139, 173–198 (1974), Zbl. 316.49005
36. Gerhardt, C. (1979): Boundary value problems for surfaces of prescribed mean curvature J. Math. Pures Appl. 58, 75–109 (1979), Zbl. 413.35024
37. Gilbarg, D., and Trudinger, N. (1983): Elliptic Partial Differential Equations of Second Order (2nd ed.), Springer-Verlag, Berlin Heidelberg New York 1983 (1st ed.: Zbl. 361.35003)
38. Giusti, E. (1972): Boundary behavior of non-parametric minimal surfaces. Indiana Univ. Math. J. 22, 435–444 (1972–73), Zbl. 262.35020
39. Giusti, E. (1976): Boundary value problems for non-parametric surfaces of prescribed mean curvature. Ann. Sc. Norm. Sup. Pisa 3, 501–548 (1976), Zbl. 344.35036
40. Giusti, E. (1984): Minimal surfaces and functions of bounded variation. Birkhäuser, Boston Basel Stuttgart 1984, Zbl. 545.49018
41. Gregori, G. (1994): Compactness and gradient bounds for solutions of the mean curvature system in two independent variables. J. Geom. Anal. 4, 327–360 (1994), Zbl. 940.54907
42. Hardt, R., Lau, C.-P., and Lin, F.-H. (1987): Nonminimality of minimal graphs. Indiana Univ. Math. J. 36, 849–855 (1987), Zbl. 637.49008
43. Heinz, E. (1952): Über die Lösungen der Minimalflächengleichung. Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. II, 51–56 (1952), Zbl. 48, 154Google Scholar
44. Hopf, E. (1950a): A theorem on the accessibility of boundary parts of an open point set. Proc. Am. Math. Soc. 1, 76–79 (1950), Zbl. 39.189
45. Hopf, E. (1950b): On S. Bernstein’s theorem on surfaces z(x, y) of non-positive curvature. Proc. Am. Math. Soc. 1, 80–85 (1950), Zbl. 39.169
46. Hopf, E. (1953): On an inequality for minimal surfaces z = z(x, y). J. Rat. Mech. Anal. 2, 519–522, 801–802 (1953), Zbl. 51, 126
47. Hildebrandt, S., Jost, J., and Widman, K. (1980): Harmonic mappings and minimal submanifolds. Invent. Math. 62, 269–298 (1980), Zbl. 446.58006
48. Huisken, G. (1989): Nonparametric mean curvature evolution with boundary conditions. J. Differ. Equations 77, 369–378 (1989), Zbl. 686.34013
49. Hwang, J.-F. (1994): Growth property for the minimal surface equation in unbounded domains. Proc. Am. Math. Soc. 121, 1027–1037 (1994), Zbl. 820.35010
50. Jenkins H. (1961a): On 2-dimensional variational problems in parametric form. Arch. Rat. Mech. Anal. 8, 181–206 (1961), Zbl. 143, 148
51. Jenkins, H. (1961b): On quasilinear equations which arise from variational problems. J. Math. Mech. 10, 705–727 (1961), Zbl. 145, 364
52. Jenkins, H., and Serrin, J. (1963): Variational problems of minimal surfaces type I. Arch. Rat. Mech. Anal. 12, 185–212 (1963), Zbl. 122, 396
53. Jenkins, H., and Serrin, J. (1968): The Dirichlet problem for the minimal surface equation in higher dimensions. J. Reine Angew. Math. 229, 170–187 (1968), Zbl. 159, 402
54. Korevaar, N. (1986): An easy proof of the interior gradient bound for solutions to the prescribed mean curvature problem. Proc. Sympos. Pure Math. 45, Part 2, 81–89 (1986), Zbl. 599.35046
55. Korevaar, N., and Simon, L. (1989): Continuity estimates for solutions to the prescribed curvature Dirichlet problem. Math. Z. 197, 457–464 (1989), Zbl. 625.35034
56. Korevaar, N., and Simon, L. (1995): Equations of mean curvature type with contact angle boundary conditions. Preprint, Stanford 1995Google Scholar
57. Korn, A. (1909): Über Minimalflächen, deren Randkurven wenig von ebenen Kurven abweichen. Berl. Abhandl. (1909), Jbuch 40, 705Google Scholar
58. Kuwert, E. (1993): On solutions of the exterior Dirichlet problem for the minimal surface equation. Ann. Inst. H. Poincaré, Anal. Non-lineaire 10, 445–451 (1993), Zbl. 820.35038
59. Ladyzhenskaya, O., and Ural’tseva, N. (1968): Linear and quasilinear elliptic equations. Academic Press, New York 1968 (translation of the original version, Moskau (1964), Zbl. 143, 336) (second Russian edition 1973)
60. Ladyzhenskaya, O., and Ural’tseva, N. (1970): Local estimates for gradients of solutions of nonuniformly elliptic and parabolic equations. Comm. Pure Appl. Math. 23, 677–703 (1970), Zbl. 193, 72
61. Langevin, R., and Rosenberg, H. (1988): A maximum principle at infinity for minimal surfaces and applications. Duke Math. J. 57, 819–826 (1988), Zbl. 667.49024
62. Lawson, H.B., and Osserman, R. (1977): Non-existence, non-uniqueness, and irregularity of solutions to the minimal surface system. Acta Math. 139, 1–17 (1977), Zbl. 376.49016
63. Leray, J., and Schauder, J. (1934): Topologie et équations fonctionelles. Ann. Sci. École Norm. Sup. 51, 45–78 (1934), Zbl. 9.73
64. Lieberman, G. (1983): The conormal derivative problem for elliptic equations of variational type. J. Differ. Equations 49, 218–257 (1983), Zbl. 506.35039
65. Lieberman, G. (1984): The nonlinear oblique derivative problem for quasilinear elliptic equations. Nonlinear Anal. 8, 49–65 (1984), Zbl. 541.35032
66. Michael, J.H., and Simon, L. (1973): Sobolev and mean-value inequalities on generalized submanifolds of ℝs n. Comm. Pure Appl. Math. 26, 361–379 (1973), Zbl. 256.53006
67. Mickle, K.J. (1950): A remark on a theorem of Serge Bernstein. Proc. Am. Math. Soc. 1, 86–89 (1950), Zbl. 39.169
68. Miranda, M. (1974): Dirichlet problem with L1 data for the non-homogeneous minimal surface equation. Indiana Univ. Math. J. 24, 227–241 (1974/5), Zbl. 293.35029
69. Miranda, M. (1977a): Superficie minime illimitate. Ann. Sc. Norm. Sup. Pisa, Ser. IV 4, 313–322 (1977), Zbl. 352.49020
70. Miranda, M. (1977b): Sulle singolarità eliminabili delle soluzioni dell’equazione delle superficie minime. Ann. Sc. Norm. Sup. Pisa, Ser. IV 4, 129–132 (1977), Zbl. 344.35037
71. Massari, U., and Miranda, M. (1984): Minimal surfaces of codimension one. Mathematics Studies 91, North Holland, Amsterdam New York 1984, Zbl. 565.49030
72. Morrey, C.B. (1938): On the solutions of quasilinear elliptic partial differential equations. Trans. Am. Math. Soc. 43, 126–166 (1938), Zbl. 18.405
73. Morrey, C.B. (1966): Multiple integrals in the calculus of variations. Springer-Verlag, Berlin-Heidelberg-New York 1966, Zbl. 142, 387
74. Moser, J. (1961): On Harnack’s theorem for elliptic differential equations. Comm. Pure Appl. Math. 14, 577–591 (1961), Zbl. 111, 93
75. Nash, J. (1958): Continuity of the solutions of parabolic and elliptic equations. Am. J. Math. 80, 931–954 (1958), Zbl. 96.69
76. Nirenberg, L. (1953): On nonlinear elliptic partial differential equations and Hölder continuity. Comm. Pure Appl. Math. 6, 103–156 (1953), Zbl. 50, 98
77. Nitsche, J.C.C. (1957): Elementary proof of Bernstein’s theorem on minimal surfaces. Ann. Math. 66, 543–544 (1957), Zbl. 79.377
78. Nitsche, J. C. C. (1965a): On new results in the theory of minimal surfaces. Bull. Am. Math. Soc. 71, 195–270 (1965), Zbl. 135, 217
79. Nitsche, J. C. C.(1965b): On the non-solvability of Dirichlet’s problem for the minimal surface equation. J. Math. Mech. 14, 779–788 (1965), Zbl. 133, 144
80. Nitsche, J.C.C. (1975): Vorlesungen über Minimalflächen. Springer-Verlag, Berlin Heidelberg New York 1975, Zbl. 319.53003
81. Nitsche, J.C.C. (1989): Lectures on Minimal Surfaces, Vol. 1. Cambridge University Press 1989, Zbl. 688.53001Google Scholar
82. Osserman, R. (1960): On the Gauss curvature of minimal surfaces. Trans. Am. Math. Soc. 96, 115–128 (1960), Zbl. 93.343
83. Osserman, R. (1973): On Bers’ theorem on isolated singularities. Indiana Univ. Math. J. 23, 337–342 (1973), Zbl. 293.53003
84. Osserman, R. (1984): The minimal surface equation. Seminar on nonlinear partial differential equations. MSRI Publications 2 (S.S. Chern, Ed.), Springer-Verlag (1984), 237–259, Zbl. 557.53033
85. Osserman, R. (1986): A Survey of Minimal Surfaces. Dover, New York 1986Google Scholar
86. Radó, T. (1930): The problem of least area and the problem of Plateau. Math. Z. 32, 763–795 (1930), Zbl. 56, 436
87. Schoen, R., Simon, L., and Yau, S.-T. (1975): Curvature estimates for minimal hypersurfaces. Acta Math. 134, 275–288 (1975), Zbl. 323.53039
88. Serrin, J. (1963): A priori estimates for solutions of the minimal surface equation. Arch. Rat. Mech. Anal. 14, 376–383 (1963), Zbl. 117, 73
89. Serrin, J. (1969): The problem of Dirichlet for quasilinear elliptic equations with many independent variables. Philos. Trans. Roy. Soc. London Ser. A 264, 413–496 (1969), Zbl. 181, 380
90. Simon, L. (1971): Interior gradient bounds for nonuniformly elliptic equations. PhD thesis, Mathematics Department, University of Adelaide 1971 (Indiana Univ. Math. J. 25, 821–855 (1976), Zbl. 346.35016)Google Scholar
91. Simon, L. (1974): Global estimates of Hölder continuity for a class of divergence form elliptic equations. Arch. Rat. Mech. Anal. 56, 253–272 (1974), Zbl. 295.35027
92. Simon, L. (1976a): Remarks on curvature estimates for minimal hypersurfaces. Duke Math. J. 43, 545–553 (1976), Zbl. 348.53003
93. Simon, L. (1976b): Interior gradient bounds for nonuniformly elliptic equations. Indiana Univ. Math. J. 25, 821–855 (1976), Zbl. 346.35016
94. Simon, L. (1976c): Boundary regularity for solutions of the non-parametric least area problem. Ann. Math. 103, 429–455 (1976), Zbl. 335.49031
95. Simon, L. (1977a): On some extensions of Bernstein’s theorem. Math. Z. 154, 265–273 (1977), Zbl. 388.49026
96. Simon, L. (1977b): A Hölder estimate for quasiconformal maps between surfaces in Euclidean space. Acta Math. 139, 19–51 (1977), Zbl. 402.30022
97. Simon, L. (1977c): Equations of mean curvature type in 2 independent variables. Pac. J. Math. 69, 245–268 (1977), Zbl. 354.35040
98. Simon, L. (1977d): On a theorem of De Giorgi and Stampacchia. Math. Z. 155, 199–204 (1977), Zbl. 385.49022
99. Simon, L. (1982): Boundary behaviour of solutions of the non-parametric least area problem. Bull. Aust. Math. Soc. 26, 17–27 (1982), Zbl. 499.49023
100. Simon, L. (1983a): Lectures on Geometric Measure Theory. Proceedings of the Centre for Mathematical Analysis, Australian National University, 3, 1983, Zbl. 546.49019Google Scholar
101. Simon, L. (1983b): Survey lectures on minimal submanifolds. In: Seminar on minimal submanifolds, Annals of Math. Studies 103 Princeton (1983), 3–52, Zbl. 541.53045Google Scholar
102. Simon, L. (1989): Entire solutions of the minimal surface equation. J. Differ. Geom. 30, 643–688 (1989), Zbl. 687.53009
103. Simon, L. (1995): Asymptotics for exterior solutions of quasilinear elliptic equations. To appear in proceedings of Pacific Rim Geometry Conference, Singapore 1995Google Scholar
104. Simon, L. (1996): Singular sets and asymptotics in geometrie analysis. Notes of Lipschitz Lectures delivered at the University of Bonn, Summer 1996Google Scholar
105. Simon, L., and Spruck J. (1976): Existence and regularity of a capillary surface with prescribd contact angle. Arch. Rat. Mech. Anal. 61, 19–34 (1976), Zbl. 361.35014
106. Simons, J. (1968): Minimal varieties in Riemannian manifolds. Ann. Math. 88, 62–105 (1968), Zbl. 181, 497
107. Spruck, J. (1974): Gauss curvature estimates for surfaces of constant mean curvature. Comm. Pure Appl. Math. 27, 547–557 (1974), Zbl. 287.53004
108. Trudinger, N. (1969a): Some existence theorems for quasilinear, non-uniformly elliptic equations in divergence form. J. Math. Mech. 18, 909–919 (1969), Zbl. 187, 357
109. Trudinger, N. (1969b): Lipschitz continuous solutions of elliptic equations of the form A(Du)D 2u = 0. Math. Z. 109, 211–216 (1969), Zbl. 174, 158
110. Trudinger, N. (1971): The boundary gradient estimate for quasilinear elliptic and parabolic differential equations. Indiana Univ. Math. J. 21, 657–670 (1971/1972), Zbl. 236.35022
111. Trudinger, N. (1972): A new proof of the interior gradient bound for the minimal surface equation in n dimensions. Proc. Nat. Acad. Sci. USA 69, 821–823 (1972), Zbl. 231.53007
112. Trudinger, N. (1973): Gradient estimates and mean curvature. Math. Z. 131, 165–175 (1973), Zbl. 253.53003
113. Ural’tseva, N. (1973): Solvability of the capillary problem. Vestn. Leningr. Univ. No. 19 (Mat. Meh. Astronom. Vyp. 4), 54–64 (1973), Zbl. 276.35045. English transl.: Vestn. Leningr. Univ., Math. 6, 363–375 (1979)Google Scholar
114. Ural’tseva, N. (1975): Sovability of the capillary problem II. Vestn. Leningr. Univ. No. 1 (Mat. Meh. Astronom. Vyp. 1 (1975), Zbl. 303.35026), English transl.: Vestn. Leningr. Univ., Math. 8, 151–158 (1980)Google Scholar
115. Williams, G. (1984): The Dirichlet problem for the minimal surface equation with Lipschitz boundary data. J. Reine Angew. Math. 354, 123–140 (1984), Zbl. 541.35033
116. Williams, G. (1986a): Solutions of the minimal surface equation –continuous and discontinuous at the boundary. Comm. Partial Differ. Equations 11, 1439–1457 (1986), Zbl. 605.49030
117. Williams, G. (1986b): Global regularity for solutions of the minimal surface equation with continuous boundary values. Ann. Inst. H. Poincaré, Anal. Non-linéaire 3, 411–429 (1986), Zbl. 627.49020
118. Yau, S.-T. (1982): Survey on partial differential equations in differential geometry. Sem. differential geometry, Ann. Math. Stud. 102 (S.-T. Yau, Ed.), Princeton University Press (1982), 3–71, Zbl. 478.53001Google Scholar

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• Leon Simon

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