Bifurcation of a capillary minimal surface in a weak gravitational field

A. Yu. Borisovich

Sbornik. Mathematics, Tome 188 (1997) no. 3, pp. 341-370

Résumé

In this paper we study a variational elliptic boundary-value problem on a convex region $\Omega \subset \mathbb R^2$ with Bond parameter $\lambda \in \mathbb R$ that arises in hydromechanics and is closely related to the Plateau problem. It describes the behaviour of an elastic surface separating two liquid or gaseous media as the gravitational field changes. In the absence of gravitational force we have $\lambda =0$ and the solution to the problem is a minimal surface. Here we study the behaviour of this surface (loss of stability, bifurcations) when gravity is introduced. The method of analysis is based on reducing the problem to an operator equation in Hölder or Sobolev spaces with a non-linear Fredholm operator of index 0 that depends on the parameter $\lambda$, and applying the Crandall–Rabinowitz theorem on simple bifurcation points, the Lyapunov–Schmidt method of reduction to finite dimensions, and the key function method due to Sapronov. We obtain both necessary and sufficient general conditions for bifurcation, and study in detail the situation when $\Omega$ is a circle or a square.

Bibliographie

[1] Dao Chong Tkhi, Fomenko A. T., Minimalnye poverkhnosti, stratifitsirovanye mnogoobraziya i problema Plato, Nauka, M., 1987 | Zbl

[2] Fomenko A. T., Variatsionnye problemy v topologii, Izd-vo MGU, M., 1984 | Zbl

[3] Dissan V. E. B., “On the spreading of liquids on solid surfaces: Static and dynamic contact lines”, Ann. Rev. Fluid Mech., 11 (1979), 371–400 | DOI

[4] Dissan V. E. B., Davis S. H., “Stability in systems with moving contact lines”, J. Fluid Mech., 173 (1986), 115–130 | DOI | MR

[5] Finn R., Equilibrium capillary surfaces, Springer-Verlag, 1986 | MR

[6] Babskii V. G., Kopachevskii N. D., Myshkis A. D., Slobozhanin L. A., Taupsov A. D., Gidromekhanika nevesomosti, Nauka, M., 1975

[7] Babskii V. G., Kopachevskii N. D., Myshkis A. D., Slobozhanin L. A., Tyuptsov A. D., “On some unsolved problems of zero dimensional hydromechanics”, Nonlinear Anal., 4 (1980), 607–621 | DOI | MR

[8] Rumyantsev V. V., Vladimirov V. A., “Teorema Lagranzha dlya tverdogo tela s polostyu, zapolnennoi idealnoi zhidkostyu”, Prikladnaya matematika i mekhanika, 53:4 (1989), 608–612 | MR | Zbl

[9] Rumyantsev V. V., Vladimirov V. A., “Teorema Lagranzha dlya tverdogo tela s polostyu, zapolnennoi vyazkoi zhidkostyu”, Prikladnaya matematika i mekhanika, 54:2 (1990), 190–200 | MR | Zbl

[10] Poston T., “A global theorem for nonlinear eigenvalue problems and applications”, Contrib. Nonlinear Fcl. Anal., Academic Press, 1971, 11–36

[11] Beeson M. J., Tromba A. J., “The cusp catastrophe of Thom in the bifurcation on minimal surfaces”, Manuscripta Math., 46:1–3 (1984), 273–308 | DOI | MR | Zbl

[12] Fomenko A. T., Tuzhilin A. A., “Mnogoznachnye otobrazheniya, minimalnye poverkhnosti i mylnye plenki”, Vestn. MGU. Ser. I, matem., mekh., 3 (1986), 3–12 | MR

[13] Tuzhilin A. A., “O bifurkatsii dvumernykh minimalnykh poverkhnostei pri dvuparametricheskoi variatsii kontura”, Geometriya, differentsialnye uravneniya, mekhanika, Izd-vo MGU, M., 1986, 140–145 | MR

[14] Buch J., “Bifurcation von Minimalflachen und elementare katastrophen”, Manuscripta Math., 55:3–4 (1986) | MR

[15] Borisovich A. Yu., “Functional-Topological Properties of the Plateau Operator and Applications to the Study of Bifurcations in Problems of Geometry and Hydrodynamics”, Adv. Soviet Math., 15 (1993), 287–330 | MR | Zbl

[16] Bernshtein S. N., Differentsialnye uravneniya s chastnymi proizvodnymi, Izd-vo AN SSSR, M., 1960

[17] Borisovich A. Yu., Zvyagin V. G., “O globalnoi obratimosti nelineinykh operatorov, porozhdaemykh kraevymi zadachami”, Priblizhennye metody issledovaniya differentsialnykh uravnenii i ikh prilozheniya, KGU, Kuibyshev, 1983, 27–33 | MR

[18] Gilbarg D., Trudinger N. S., Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1983 | MR | Zbl

[19] Kondratev V. A., “Kraevye zadachi dlya ellipticheskikh uravnenii v oblastyakh s konicheskimi i uglovymi tochkami”, Tr. MMO, 16, URSS, M., 1967, 209–293

[20] Crandall M. G., Rabinowitz P. H., “Bifurcation from simple eigenvalues”, Funct. Anal., 8 (1971), 321–340 | DOI | MR | Zbl

[21] Nirenberg L., Topics in nonlinear functional analysis, Courant Inst. of Math. Sci., New York, 1974 | MR | Zbl

[22] Krasnoselskii M. A., Topologicheskie metody v teorii nelineinykh integralnykh uravnenii, Gostekhizdat, M., 1956 | MR

[23] Rabinovich P. H., “The Problem of Plateau, an introduction to the whole Mathematics”, Summer conference (Trieste), Mimeographical Notes, 1972

[24] Ize I. A., “Bifurcation theory for Fredholm operators”, Mem. Amer. Math. Soc., 174 (1976) | MR | Zbl

[25] Arnold V. I., Gusein-Zade S. M., Varchenko A. N., Osobennosti differentsiruemykh otobrazhenii, T. I, II, Nauka, M., 1982 | MR

[26] Sapronov Yu. I., “Branching of solutions of smooth Fredholm equations”, Lecture Notes in Math., 1108, 1982, 31–54 | MR

[27] Vainberg M. M., Trenogin V. A., Teoriya vetvleniya reshenii nelineinykh uravnenii, Nauka, M., 1969 | MR