Hostname: page-component-745bb68f8f-l4dxg Total loading time: 0 Render date: 2025-02-05T08:21:35.396Z Has data issue: false hasContentIssue false
  • English
  • Français

On non-linear elliptic equations and the stability of soap film

Published online by Cambridge University Press:  17 April 2009

Công Nghê Truong
Công Nghê Truong
Affiliation:
School of Quantitative Business Methods and OperationsUniversity of Western Sydney, NepeanPO Box 10Kingswood NSW 2747Australia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We use an Inverse Mapping Theorem for Sobolev chains to obtain a number of existence theorems for non-linear elliptic partial differential equations with general order. We then apply one of these theorems to prove the stability of a soap film.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 57 , Issue 3 , June 1998 , pp. 353 - 361
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Agmon, S., Douglas, A. and Nirenberg, L., ‘Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions’, Comm. Pure Appl. Math. XII (1959), 623–727.CrossRefGoogle Scholar
[2]Bateman, H., Partial differential equations of mathematical physics (Cambridge University Press, Cambridge, England, 1959).Google Scholar
[3]Bers, L. and Schechter, M., Partial differential equations, Lectures in Applied Mathematics III (John Wiley and Sons, New York, 1964).Google Scholar
[4]Browder, F.E., ‘Topological methods for non-linear elliptic equations of arbitrary order’, Pacific J. Math. 17 (1966), 17–31.CrossRefGoogle Scholar
[5]Browder, F.E., ‘Existence theorems for nonlinear partial differential equations’, in Global Analysis, Proceedings of Symposia in Pure Mathematics XVI (American Mathematical Society, Providence, R.I., 1970).Google Scholar
[6]Courant, R., ‘Soap film experiments with minimal surfaces’, Amer. Math. Monthly 47 (1940), 167–174.CrossRefGoogle Scholar
[7]Courant, R., Dirichlet's principle, conformal mapping and minimal surfaces (Interscience, New York, 1950).Google Scholar
[8]Courant, R. and Hilbert, D., Methods of Mathematical Physics 2 (Interscience, New York, 1961).Google Scholar
[9]Eels, J., ‘A setting for global analysis’, Bull. Amer. Math. Soc. 72 (1966), 751–807.CrossRefGoogle Scholar
[10]Garabedian, P.R., Partial differential equations (John Wiley and Sons, New York, 1964).Google Scholar
[11]Hormander, L., Linear partial differential operators, (Third revised printing) (Springer-Verlag, Berlin, Heidelberg, New York, 1969).CrossRefGoogle Scholar
[12]Marsden, J.E., Hamiltonian mechanics, infinite dimensional Lie groups, Geodesic flows and hydrodynamics, (Notes at Berkeley, California, 1969).Google Scholar
[13]Nirenberg, L., ‘On elliptic partial differential equations’, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 13 (1969), 115–162.Google Scholar
[14]Omori, H., Infinite dimensional Lie transformation groups, Lecture Notes in Mathematics 427 (Springer-Verlag, Berlin, Heidelberg, New York, 1974).CrossRefGoogle Scholar
[15]Smale, S., ‘An infinite dimensional version of Sard's Theorem’, Amer. J. Math. 87 (1965), 861–866.CrossRefGoogle Scholar
[16]Sunada, T., ‘Non-linear elliptic operators on a compact manifold and an implicit function theorem’, Nagoya Math. J. 57 (1974), 175–200.Google Scholar
[17]Truong, C.N., ‘Manifolds of smooth maps’, Bull. Austral. Math. Soc. 24 (1981), 1–11.Google Scholar
[18]Truong, C.N., ‘An inverse mapping theorem for Sobolev chains and its application’, Bull. Austral. Math. Soc. 27 (1983), 281–394.Google Scholar
[19]Truong, C.N. and Yamamuro, S., ‘Locally convex spaces, differentiation and manifolds’, Comment. Math. Special Issue 2 (1979), 229–338.Google Scholar
[20]Yamamuro, S., A theory of differentiation in locally convex spaces, Memoirs of the Arnerical Mathematical Society 212 (American Mathematical Society, Providence, RI, 1979).Google Scholar