Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

Boundary value problems for variational integrals involving surface curvatures


Author: Johannes C. C. Nitsche
Journal: Quart. Appl. Math. 51 (1993), 363-387
MSC: Primary 58E12; Secondary 35J55, 53A10, 76B45
DOI: https://doi.org/10.1090/qam/1218374
MathSciNet review: MR1218374
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The following investigation deals with surfaces governed by and extremal for a free energy functional which is quadratic in the principal curvatures. The associated Euler-Lagrange differential equations are derived, as are the corresponding intricate natural boundary conditions. Pertinent boundary value problems--without and with volume constraints--are formulated and discussed$ ^{1}$ and existence proofs are provided for certain situations. The discussion opens the view onto an arena of rich mathematical problems which will also be of interest in engineering applications where the surfaces in question are utilized frequently as idealized models for the interfaces separating phases in real materials.


References [Enhancements On Off] (What's this?)

  • [1] D. M. Anderson, Studies in the microstructure of microemulsion, Dissertation, Univ. of Minnesota, 1986
  • [2] D. M. Anderson, H. T. Davis, J. C. C. Nitsche, and L. E. Scriven, Periodic surfaces of prescribed mean curvature, Advances in Chem. Phys., vol. 77, Interscience, New York, 1990, pp. 337-396
  • [3] D. M. Anderson, S. M. Gruner, and S. Leibler, Geometric aspects of the frustration in the cubic phases of lyotropic liquid crystals, Proc. Nat. Acad. Sci. U.S.A. 85, 5364-5368 (1988)
  • [4] S. Balasundaram and P. K. Bhattacharyya, On existence of solution of the Dirichlet problem of fourth order partial differential equations with variable coefficients, Quart. Appl. Math. 41, 311-317 (1983) MR 721421
  • [5] S. N. Bernstein, Sur les surfaces définies au moyen de leur courbure moyenne et totale, Ann. Sci. École Norm. Sup. 27, 233-256 (1910)
  • [6] R. Bryant, A duality theorem for Willmore surfaces, J. Differential Geom. 20, 23-53 (1984) MR 772125
  • [7] F. Casorati, Mesure de la courbure des surfaces suivant l'idée commune. Ses rapports avec les mesures de courbure Gaussienne et moyenne, Acta Math. 14, 95-110 (1890/91) MR 1554792
  • [8] B. Y. Chen, An invariant of conformal mappings, Proc. Amer. Math. Soc. 40, 563-564 (1973) MR 0320956
  • [9] B. Y. Chen, Some conformal invariants of submanifolds and their applications, Boll. Un. Mat. Ital. (4) 10, 380-385 (1974) MR 0370436
  • [10] B. Y. Chen, On a variational problem on hypersurfaces, J. London Math. Soc. (2) 6, 321-325 (1973) MR 0312437
  • [11] H. T. Davis and L. E. Scriven, Stress and structure in fluid interfaces, Advances in Chem. Phys. 49, 357-454 (1982)
  • [12] G. Dziuk, Über quasilineare Systeme mit isothermen Parametern an Ecken der Randkurven, Analysis 1, 63-81 (1981) MR 623643
  • [13] G. Dziuk, Über die Stetigkeit teilweise freier Minimalflächen, Manuscripta Math. 36, 241-251 (1981) MR 641976
  • [14] W. Fenchel, Über Krümmung und Windung geschlossener Raumkurven, Math. Ann. 101, 238-252 (1929) MR 1512528
  • [15] R. Finn, Remarks relevant to minimal surfaces and to surfaces of prescribed mean curvature, J. Analyse Math. 14, 139-160 (1965) MR 0188909
  • [16] W. Fischer and E. Koch, New surface patches for minimal balance surfaces. I, Branched catenoids, Acta Cryst. A 45, 166-169 (1989); II, Multiple catenoids, Acta Cryst. A 45, 169-174 (1989); III, Infinite strips, Acta Cryst. A 45, 485-490 (1989); IV, Catenoids with spout-like attachments, Acta Cryst. A 45, 558-563 (1989) MR 983425
  • [17] A. Fogden, S. T. Hyde, and G. Lundberg, Bending energy of surfactant films, J. Chem. Soc. Faraday Trans. 87 (7), 949-955 (1991)
  • [18] P. Funk, Variationsrechnung und ihre Anwendung in Physik und Technik, 2nd ed., Springer-Verlag, Berlin, Heidelberg, and New York, 1970 MR 0276847
  • [19] C. Gerhardt, Existence, regularity, and boundary behaviour of generalized surfaces of prescribed mean curvature, Math. Z. 139, 173-198 (1974) MR 0437925
  • [20] G. Gerhardt, Boundary value problems for surfaces of prescribed mean curvature, J. Math. Pures Appl. 58, 75-109 (1979) MR 533236
  • [21] S. Germain, Recherches sur la théorie des surfaces élastiques, Imprimerie de Huzard-Courcier, Paris, 1921
  • [22] M. Giaquinta, Regolarità delle superfici BV $ BV\left( \Omega \right)$ con curvatura media assegnata, Boll. Un. Mat. Ital. 8, 567-578 (1973) MR 0377669
  • [23] M. Giaquinta, On the Dirichlet problem for surfaces of prescribed mean curvature, Manuscripta Math. 12, 73-86 (1974) MR 0336532
  • [24] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Springer-Verlag, Berlin, Heidelberg, New York, and Tokyo, 1983 MR 737190
  • [25] E. Giusti, Boundary value problems for non-parametric surfaces of prescribed mean curvature, Ann. Scuola Norm. Sup. Pisa (4) 3, 501-548 (1976) MR 0482506
  • [26] E. Giusti, On the equation of surfaces of prescribed mean curvature, Invent. Math. 46, 111-137 (1978) MR 0487722
  • [27] P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Math., vol. 24, Pitman Advanced Pub. Program, Boston, 1985 MR 775683
  • [28] S. M. Gruner, Stability of lyotropic phases with curved interfaces, J. Phys. Chem. 93, 7562-7570 (1989)
  • [29] M. Grüter, S. Hildebrandt, and J. C. C. Nitsche, On the boundary behavior of minimal surfaces with a free boundary which are not minima of the area, Manuscripta Math. 35, 387-410 (1981) MR 636464
  • [30] M. Grüter, S. Hildebrandt, and J. C. C. Nitsche, Regularity for surfaces of constant mean curvature with free boundaries, Acta Math. 156, 119-152 (1986) MR 822332
  • [31] E. Heinz, Über Flächen mit eineindeutiger Projektion auf eine Ebene, deren Krümmungen durch Ungleichungen eingeschränkt sind, Math. Ann. 129, 451-454 (1955) MR 0071822
  • [32] E. Heinz, On the nonexistence of a surface of constant mean curvature with finite area and prescribed rectifiable boundary, Arch. Rational Mech. Anal. 35, 249-252 (1969) MR 0246237
  • [33] E. Heinz, Minimalflächen mit polygonalem Rand, Math. Z. 183, 547-564 (1983) MR 710771
  • [34] W. Helfrich, Elastic properties of lipid bilayers: Theory and possible experiments, Z. Naturforsch. A 28c, 693-703 (1973)
  • [35] W. Helfrich and H. Rennschuh, Landau theory of the lamellar-to-cubic phase transition, Colloque de Physique C7, 51, supp. au no. 23, 189-195 (1990)
  • [36] S. Hildebrandt and J. C. C. Nitsche, A uniqueness theorem for surfaces of least area with partially free boundaries on obstacles, Arch. Rational Mech. Anal. 79, 189-218 (1982) MR 658387
  • [37] S. T. Hyde, Curvature and the global structure of interfaces in surfactant-water systems, Colloque de Physique C7, 51, suppl. au no. 23, 207-228 (1990) MR 1090150
  • [38] S. T. Hyde, I. S. Barnes, and B. W. Ninham, Curvature energy of surfactant interfaces confined to the plaquettes of a cubic lattice, Langmuir, ACS J. Surf. Colloids 6, 1055-1062 (1990)
  • [39] J. N. Israelachvili, Intermolecular and surface forces: With applications to colloidal and biological systems, Academic Press, London and Orlando, 1985
  • [40] J. N. Israelachvili and P. M. McGuiggan, Forces between surfaces in liquids, Science 241, 795-800 (1988)
  • [41] N. Kapouleas, Complete constant mean curvature surfaces in euclidean three-space, Ann. of Math. 131, 239-330 (1990) MR 1043269
  • [42] H. Karcher, The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions, Manuscripta Math 64, 291-337 (1989) MR 1003093
  • [43] H. Karcher, U. Pinkall, and I. Sterling, New minimal surfaces in $ S^{3}$ , J. Differential Geom. 28, 169-185 (1988) MR 961512
  • [44] G. R. Kirchhoff, Über das Gleichgewicht und die Bewegung einer elastischen Scheibe, J. Reine Angew. Math. 40, 51-88 (1850)
  • [45] K. L. Kirk, S. M. Gruner, and D. L. Stein, Thermodynamic model of the lamellar to inverse hexagonal phase transition of lipid membrane-water systems, Biochem. 23, 1093-1102 (1984)
  • [46] V. A. Kondrat'ev and O. A. Oleinik, Boundary value problems for partial differential equations in non-smooth domains, Russian Math. Surv. 38, 1-86 (1983) MR 695471
  • [47] H. B. Lawson, Complete minimal surfaces in $ S^{3}$ , Ann. of Math. (2) 92, 335-374 (1970) MR 0270280
  • [48] A. E. H. Love, A treatise on the mathematical theory of elasticity, 2nd ed., Cambridge University Press, Cambridge, 1906
  • [49] H. Minkowski, Kapillarität, Encyclopedia Math. Wissen. 5.1.9, B. G. Teubner, Leipzig, 1903-1921 (completed 1906), pp. 558-613
  • [50] J. C. C. Nitsche, Concerning the isolated character of solutions of Plateau's problem, Math. Z. 109, 393-411 (1969) MR 0243439
  • [51] J. C. C. Nitsche, Vorlesungen über Minimalflächen, Springer-Verlag, Berlin, Heidelberg, and New York, 1975 MR 0448224
  • [52] J. C. C. Nitsche, A volume formula, Analysis 3, 337-346 (1983) MR 756122
  • [53] J. C. C. Nitsche, Stationary partitioning of convex bodies, Arch. Rational Mech. Anal. 89, 1-19 (1985); corrigendum in Arch. Rational Mech. Anal. 95, 389 (1986) MR 784101
  • [54] J. C. C. Nitsche, Lectures on minimal surfaces, Cambridge University Press, Cambridge, New York, New Rochelle, Melbourne, and Sydney, 1989 MR 1015936
  • [55] J. C. C. Nitsche, Periodic surfaces which are extremal for energy functional containing curvature functions, IMA Preprint no. 785, March 1991 (to appear in Proc. Workshop Statistical Thermodynamics and Differential Geometry of Microstructured Materials, H. T. Davis and J. C. C. Nitsche, eds., IMA vol. in Math. and its Appl.) MR 1226921
  • [56] J. C. C. Nitsche, Mathematik in Berlin, Born konkreter Geometrie über die Jahrhunderte, Wissenschaft und Stadt (D. Heckelmann and O. Büsche, eds.), Colloquium Verlag, Berlin (to appear) MR 1607410
  • [57] Ou-Yang Zhong-can, Anchor ring vesicle membranes, Phys. Rev. A 41, 4517-4520 (1990)
  • [58] Ou-Yang Zhong-can and W. Helfrich, Bending energy of vesicle membranes: General expressions for the first, second, and third variation of the shape energy and application to spheres and cylinders, Phys. Rev. A 39, 5280-5288 (1989)
  • [59] S. D. Poisson, Mémoire sur les surfaces élastiques, Cl. Sci. Mathém. Phys. Inst. de France, 2nd pt., 167-225 (1812)
  • [60] S. D. Poisson, Mémoire sur le calcul des variations, Mém. Acad. Roy. Sci. Inst. de France 12, 223-331 (1833)
  • [61] E. Reissner, The effect of transverse shear deformation on the bending of the elastic plates, J. Appl. Mech. 12, A68-A77 (1945) MR 0012579
  • [62] E. Reissner, On bending of elastic plates, Quart. Appl. Math. 5, 55-68 (1947) MR 0020440
  • [63] J. S. Rowlinson and B. Widom, Molecular theory of capillarity, Clarendon Press, Oxford, 1982
  • [64] M. Schäfer, Über eine Verfeinerung der klassischen Theorie dünner schwach gebogener Platten, Z. Angew. Math. Mech. 32, 161-171 (1952) MR 0053729
  • [65] A. H. Schoen, Infinite periodic minimal surfaces without self-intersections, NASA Techn. Rep. D-5541 (1970)
  • [66] H. A. Schwarz, Gesammelte Mathematische Abhandlungen, vol. 1, Springer, Berlin, 1890
  • [67] H. A. Schwarz, Zur Theorie der Minimalflächen, deren Begrenzung aus geradlinigen Strecken besteht, Sitz.-Ber. Königl. Preuss., Akad. d. Wiss. Berlin, Phys.-Math. Cl., 1237-1266 (1894)
  • [68] L. E. Scriven, Equilibrium bicontinuous structure, Nature 263, 123-125 (1976)
  • [69] J. Serrin, The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables, Philos. Trans. Roy. Soc. London Ser. A 264, 413-496 (1969) MR 0282058
  • [70] L. Simon, Existence of Willmore surfaces, Proc. Centre Math. Anal., Austra. Nat. Univ. 10, 187-216 (1985) MR 857667
  • [71] I. Sterling, Willmore surfaces and computers, (to appear in Proc. Workshop Statistical Thermodynamics and Differential Geometry of Microstructured Materials, H. T. Davis and J. C. C. Nitsche, eds., IMA vol. in Math. and its Appl.) MR 1226925
  • [72] G. Thomsen, Über konforme Geometrie. I, Grundlagen der konformen Flächentheorie, Abh. Math. Sem. Univ. Hamburg 3, 31-56 (1924) MR 3069418
  • [73] J. L. Weiner, On a problem of Chen, Willmore, et al., Indiana Univ. Math. J. 27, 19-35 (1978) MR 0467610
  • [74] J. H. White, A global invariant of conformal mappings in space, Proc. Amer. Math. Soc. 38, 162-164 (1973) MR 0324603
  • [75] N. M. Wigley, Mixed boundary problems in plane domains with corners, Math. Z. 115, 33-52 (1970) MR 0262669
  • [76] N. M. Wigley, Schauder estimates in domains with corners, Arch. Rational Mech. Anal. 104, 271-276 (1988) MR 1017291
  • [77] T. J. Willmore, Total curvature in Riemannian geometry, Ellis Horwood Ltd., Chichester, 1982 MR 686105

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 58E12, 35J55, 53A10, 76B45

Retrieve articles in all journals with MSC: 58E12, 35J55, 53A10, 76B45


Additional Information

DOI: https://doi.org/10.1090/qam/1218374
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society