Abstract
The minimal surface equation (MSE) for functions u: Ω → ℝ, Ω a domain of ℝ2, can be written
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 ⊂ ℝn and functions Ω → ℝ depending on the n variables (x 1, …, x n) ∈ Ω, n ≥ 2, the MSE can be written
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”
which is readily checked using the chain rule and the fact that \(\frac{\partial }{{\partial {p_j}}}\left( {\frac{p}{{\sqrt {1 + |p{|^2}} }}} \right) = {\left( {1 + |p{|^2}} \right)^{ - 1/2}}\left( {{\delta _{ij}} - \frac{{{p_i}{p_j}}}{{1 + |p{|^2}}}} \right)\).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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
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
Aronsson, G. (1968): On the partial differential equation (math). Ark. Mat. 7, 395–425 (1968), 162, 422
Bers, L. (1951): Isolated singularities of minimal surfaces. Ann. Math. 53, 364–386 (1951), Zbl. 43, 159
Bers, L. (1954): Non-linear elliptic equations without non-linear entire solutions. J. Rat. Mech. Anal. 3. 767–787 (1954), Zbl. 56, 321
Bernstein, S. (1910): Sur la généralisation du problème Dirichlet II. Math. Ann. 69, 82–136 (1910), Jbuch 41, 427
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))
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
Bombieri, E., De Giorgi, E., and Giusti, E. (1969): Minimal cones and the Bernstein problem. Invent. Math. 7, 243–268 (1969), Zbl. 183, 259
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
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
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
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. 668.35028
Courant, R., and Hilbert, D. (1962): Methods of Mathematical Physics, Vol. II. Interscience Publishers, New York 1962, Zbl. 99, 295
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. 84.319
De Giorgi, E. (1961): Frontiere orientate di misura minima. Sem. Mat. Sc. Norm. Sup. Pisa, 1–56 (1961)
De Giorgi, E. (1965): Una estensione del teorema di Bernstein. Ann. Sc. Norm. Sup. Pisa 19, 79–85 (1965), Zbl. 168, 98
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
Dierkes, U. (1993): A Bernstein result for energy minimizing hypersurfaces. Calc. Var. Partial Differ. Equ. 1, 37–54 (1993), Zbl. 819.35030
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
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
Ecker, K., and Huisken, G. (1989): Mean curvature evolution of entire graphs. Ann. Math. 130, 453–471 (1989), Zbl. 696.53036
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
Ecker, K., and Huisken, G. (1991): Interior estimates for hypersurfaces moving by mean curvature. Invent. Math. 105, 547–569 (1991), Zbl. 725.53009
Federer, H. (1969): Geometric Measure Theory. Springer-Verlag, Berlin Heidelberg New York 1969, Zbl. 176, 8
Finn, R. (1953): A property of minimal surfaces. Proc. Nat. Acad. Sci. USA 39, 197–201 (1953), Zbl. 51, 125
Finn, R. (1954): On equations of minimal surface type. Ann. Math. 60, 397–416 (1954), Zbl. 58, 325
Finn, R. (1963): New estimates for equations of minimal surface type. Arch. Rat. Mech. Anal. 14, 337–375 (1963), Zbl. 133, 46
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
Finn, R. (1986): Equilibrium Capillary Surfaces. Springer-Verlag, Berlin Heidelberg New York 1986, Zbl. 583.35002
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
Finn, R., and Osserman, R. (1964): The gauss curvature of nonparametric minimal surfaces. J. Anal. Math. 12, 351–364 (1964), Zbl. 122, 164
Fischer-Colbrie, D. (1980): Some rigidity theorems for minimal submanifolds of the sphere. Acta. Math. 145, 29–46 (1980), Zbl. 464.53047
Fleming, W. (1962): On the oriented Plateau problem. Rend. Circ. Mat. Palermo 11, 69–90 (1962), Zbl. 107, 313
Gerhardt, C. (1974): Existence, regularity, and boundary behaviour of generalized surfaces of prescribed mean curvature. Math. Z. 139, 173–198 (1974), Zbl. 316.49005
Gerhardt, C. (1979): Boundary value problems for surfaces of prescribed mean curvature J. Math. Pures Appl. 58, 75–109 (1979), Zbl. 413.35024
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)
Giusti, E. (1972): Boundary behavior of non-parametric minimal surfaces. Indiana Univ. Math. J. 22, 435–444 (1972–73), Zbl. 262.35020
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
Giusti, E. (1984): Minimal surfaces and functions of bounded variation. Birkhäuser, Boston Basel Stuttgart 1984, Zbl. 545.49018
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
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
Heinz, E. (1952): Über die Lösungen der Minimalflächengleichung. Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. II, 51–56 (1952), Zbl. 48, 154
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
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
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
Hildebrandt, S., Jost, J., and Widman, K. (1980): Harmonic mappings and minimal submanifolds. Invent. Math. 62, 269–298 (1980), Zbl. 446.58006
Huisken, G. (1989): Nonparametric mean curvature evolution with boundary conditions. J. Differ. Equations 77, 369–378 (1989), Zbl. 686.34013
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
Jenkins H. (1961a): On 2-dimensional variational problems in parametric form. Arch. Rat. Mech. Anal. 8, 181–206 (1961), Zbl. 143, 148
Jenkins, H. (1961b): On quasilinear equations which arise from variational problems. J. Math. Mech. 10, 705–727 (1961), Zbl. 145, 364
Jenkins, H., and Serrin, J. (1963): Variational problems of minimal surfaces type I. Arch. Rat. Mech. Anal. 12, 185–212 (1963), Zbl. 122, 396
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
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
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
Korevaar, N., and Simon, L. (1995): Equations of mean curvature type with contact angle boundary conditions. Preprint, Stanford 1995
Korn, A. (1909): Über Minimalflächen, deren Randkurven wenig von ebenen Kurven abweichen. Berl. Abhandl. (1909), Jbuch 40, 705
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
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)
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
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
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
Leray, J., and Schauder, J. (1934): Topologie et équations fonctionelles. Ann. Sci. École Norm. Sup. 51, 45–78 (1934), Zbl. 9.73
Lieberman, G. (1983): The conormal derivative problem for elliptic equations of variational type. J. Differ. Equations 49, 218–257 (1983), Zbl. 506.35039
Lieberman, G. (1984): The nonlinear oblique derivative problem for quasilinear elliptic equations. Nonlinear Anal. 8, 49–65 (1984), Zbl. 541.35032
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
Mickle, K.J. (1950): A remark on a theorem of Serge Bernstein. Proc. Am. Math. Soc. 1, 86–89 (1950), Zbl. 39.169
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
Miranda, M. (1977a): Superficie minime illimitate. Ann. Sc. Norm. Sup. Pisa, Ser. IV 4, 313–322 (1977), Zbl. 352.49020
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
Massari, U., and Miranda, M. (1984): Minimal surfaces of codimension one. Mathematics Studies 91, North Holland, Amsterdam New York 1984, Zbl. 565.49030
Morrey, C.B. (1938): On the solutions of quasilinear elliptic partial differential equations. Trans. Am. Math. Soc. 43, 126–166 (1938), Zbl. 18.405
Morrey, C.B. (1966): Multiple integrals in the calculus of variations. Springer-Verlag, Berlin-Heidelberg-New York 1966, Zbl. 142, 387
Moser, J. (1961): On Harnack’s theorem for elliptic differential equations. Comm. Pure Appl. Math. 14, 577–591 (1961), Zbl. 111, 93
Nash, J. (1958): Continuity of the solutions of parabolic and elliptic equations. Am. J. Math. 80, 931–954 (1958), Zbl. 96.69
Nirenberg, L. (1953): On nonlinear elliptic partial differential equations and Hölder continuity. Comm. Pure Appl. Math. 6, 103–156 (1953), Zbl. 50, 98
Nitsche, J.C.C. (1957): Elementary proof of Bernstein’s theorem on minimal surfaces. Ann. Math. 66, 543–544 (1957), Zbl. 79.377
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
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
Nitsche, J.C.C. (1975): Vorlesungen über Minimalflächen. Springer-Verlag, Berlin Heidelberg New York 1975, Zbl. 319.53003
Nitsche, J.C.C. (1989): Lectures on Minimal Surfaces, Vol. 1. Cambridge University Press 1989, Zbl. 688.53001
Osserman, R. (1960): On the Gauss curvature of minimal surfaces. Trans. Am. Math. Soc. 96, 115–128 (1960), Zbl. 93.343
Osserman, R. (1973): On Bers’ theorem on isolated singularities. Indiana Univ. Math. J. 23, 337–342 (1973), Zbl. 293.53003
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
Osserman, R. (1986): A Survey of Minimal Surfaces. Dover, New York 1986
Radó, T. (1930): The problem of least area and the problem of Plateau. Math. Z. 32, 763–795 (1930), Zbl. 56, 436
Schoen, R., Simon, L., and Yau, S.-T. (1975): Curvature estimates for minimal hypersurfaces. Acta Math. 134, 275–288 (1975), Zbl. 323.53039
Serrin, J. (1963): A priori estimates for solutions of the minimal surface equation. Arch. Rat. Mech. Anal. 14, 376–383 (1963), Zbl. 117, 73
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
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)
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
Simon, L. (1976a): Remarks on curvature estimates for minimal hypersurfaces. Duke Math. J. 43, 545–553 (1976), Zbl. 348.53003
Simon, L. (1976b): Interior gradient bounds for nonuniformly elliptic equations. Indiana Univ. Math. J. 25, 821–855 (1976), Zbl. 346.35016
Simon, L. (1976c): Boundary regularity for solutions of the non-parametric least area problem. Ann. Math. 103, 429–455 (1976), Zbl. 335.49031
Simon, L. (1977a): On some extensions of Bernstein’s theorem. Math. Z. 154, 265–273 (1977), Zbl. 388.49026
Simon, L. (1977b): A Hölder estimate for quasiconformal maps between surfaces in Euclidean space. Acta Math. 139, 19–51 (1977), Zbl. 402.30022
Simon, L. (1977c): Equations of mean curvature type in 2 independent variables. Pac. J. Math. 69, 245–268 (1977), Zbl. 354.35040
Simon, L. (1977d): On a theorem of De Giorgi and Stampacchia. Math. Z. 155, 199–204 (1977), Zbl. 385.49022
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
Simon, L. (1983a): Lectures on Geometric Measure Theory. Proceedings of the Centre for Mathematical Analysis, Australian National University, 3, 1983, Zbl. 546.49019
Simon, L. (1983b): Survey lectures on minimal submanifolds. In: Seminar on minimal submanifolds, Annals of Math. Studies 103 Princeton (1983), 3–52, Zbl. 541.53045
Simon, L. (1989): Entire solutions of the minimal surface equation. J. Differ. Geom. 30, 643–688 (1989), Zbl. 687.53009
Simon, L. (1995): Asymptotics for exterior solutions of quasilinear elliptic equations. To appear in proceedings of Pacific Rim Geometry Conference, Singapore 1995
Simon, L. (1996): Singular sets and asymptotics in geometrie analysis. Notes of Lipschitz Lectures delivered at the University of Bonn, Summer 1996
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
Simons, J. (1968): Minimal varieties in Riemannian manifolds. Ann. Math. 88, 62–105 (1968), Zbl. 181, 497
Spruck, J. (1974): Gauss curvature estimates for surfaces of constant mean curvature. Comm. Pure Appl. Math. 27, 547–557 (1974), Zbl. 287.53004
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
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
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
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
Trudinger, N. (1973): Gradient estimates and mean curvature. Math. Z. 131, 165–175 (1973), Zbl. 253.53003
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)
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)
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
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
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
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. 478.53001
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Simon, L. (1997). The Minimal Surface Equation. In: Osserman, R. (eds) Geometry V. Encyclopaedia of Mathematical Sciences, vol 90. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03484-2_5
Download citation
DOI: https://doi.org/10.1007/978-3-662-03484-2_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-08225-2
Online ISBN: 978-3-662-03484-2
eBook Packages: Springer Book Archive