Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-23T23:21:14.947Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-linear elliptic operators on a compact manifold and an implicit function theorem

Published online by Cambridge University Press:  22 January 2016

Toshikazu Sunada
Toshikazu Sunada*
Affiliation:
Nagoya University
Rights & Permissions [Opens in a new window]

Extract

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.

Many problems in differential and analytic geometry seem to have something to do with the study of non-linear partial differential equations of elliptic type. For instance, the classical Weyl and Minkowski problems for a convex surface have been studied by H. Weyl, H. Lewy, and L. Nirenberg using the iteration method for the construction of solutions of certain non-linear equations of elliptic type (see [6]). Also, M. Kuranishi [3] constructed the effective complete family of deformations of complex analytic structures on a given compact complex manifold as the solution space of another non-linear equation of elliptic type; whereby the basic idea in his work is to apply an implicit function theorem to the non-linear operator of a Banach space, and to construct the bifurcation of solutions explicitly.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 56 , January 1975 , pp. 175 - 200
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1975

References

[1] Douglis, A. and Nirenberg, L.: Interior estimate for elliptic systems of partial differential equations, Comm. Pure Appl. Math., 8 (1955), 503–538.CrossRefGoogle Scholar
[2] Eells, J.: A setting for Global Analysis, Bull. Amer. Math. Soc., 72 (1966), 751–809.CrossRefGoogle Scholar
[3] Kuranishi, M.: New proof for existence of locally complete families of complex structure, Proceeding of the Conference on Complex Analysis, Minneapolis (1964), 142–154.Google Scholar
[4] Moser, J.: A new technique for the construction of solutions of non-linear differential equations, Proc. Nut. Acad. U.S.A., 47 (1961), 1824–1831.CrossRefGoogle Scholar
[5] Nash, J.: The imbedding problem for Riemannian manifolds, Ann. Math., 63 (1956), 20–63.CrossRefGoogle Scholar
[6] Nirenberg, L.: The Weyl and Minkowski problems in differential geometry in the large, Comm. Pure Appl. Math., 6 (1953), 337–394.CrossRefGoogle Scholar
[7] Ômori, H.: On the group of diffeomorphisms on a compact manifold, Proc. Symp. Pure Math., Vol. XV, Amer. Math. Soc. (1970), 167–183.CrossRefGoogle Scholar
[8] Palais, R. S.: Seminar on the Atiyah Singer index theorem, Princeton study 57 (1965).CrossRefGoogle Scholar
[9] Palais, R. S.: Foundation of Global Non-linear Analysis, W. A. Benjamin Inc. (1968).Google Scholar
[10] Sunada, T.: The solution spaces of non-linear partial differential equations of elliptic type on compact manifolds, Proc. Japan. Acad., 49 (1973 (, 385–389.Google Scholar