Article PDF
Avoid common mistakes on your manuscript.
Bibliography
Ahlfors, L. V., Conformality with respect to Riemannian metrics,Annales Academiae Scientiarum Fennicae, Series A. I. Mathematica-Physica, No. 206 (1955).
Alexandrow, A. D., Smoothness of the convex surface of bounded Gaussian curvature,Doklady. Acad. Nauk. U.S.S.R. (N.S.) 36, pp. 195–199 (1942).
Bernstein, S., Sur les surfaces définies au moyen de leur courbure moyenne ou totale,Ann. Ecole. Norm. Sup. 27, pp. 233–256 (1910).
Bianchi, L., Vorlesungen über Differentialgeometrie, Leipzig und Berlin, 1910.
Blaschke, W., Kreis und Kugel, Leipzig, 1916.
Courant, R. and Hilbert, D., Methoden der mathematischen Physik, 11, Berlin, 1937.
Darboux, G., Théorie générale des surfaces, Vol. 3, Paris, 1894.
Douglis, A. and Nirenberg, L., Interior estimates for elliptic systems of partial differential equations,Communications on Pure and Applied Mathematics, Vol. VIII, pp. 503–538 (1955).
Efimow, N. V., Qualitative problems of the theory of deformation of surfaces,Uspehi Mat. Nauk, (N.S.), No., 2 (24), pp. 47–158 (1948) (Am. Math. Soc. Translation Number 37).
Hadamard, J., Le problème de Cauchy et les équations aux dérivées partielles linéaires hyperboliques, Paris, 1932.
Heinz, E., Ueber gewisse elliptische Systeme von Differentialgleichungen zweiter Ordnung mit Anwendung auf die Monge-Ampèresche Gleichung,Mathematische Annalen, Vol. 131, pp. 411–428 (1956).
Heinz, E., On certain non-linear elliptic differential equations and univalent mappings,Journal d'Analyse Mathématique, Vol. V, pp. 197–272 (1956/57).
Hopf, E., Ueber den funktionalen, insbesondere den analytischen Charakter der Lösungen elliptischer Differentialgleichungen zweiter Ordnung,Math. Zeitschrift, Vol. 34, No. 2, pp. 191–233 (1931).
Korn, A., Zwei Anwendungen der Methode der sukzessiven Annäherung,Schwarz-Festschrift, pp. 215–229, Berlin, 1914.
Leray, J., Discussion d'un problème de Dirichlet,Journal de Mathématiques Pures et Appliquées, Vol. 18, No. 9, pp. 249–284 (1939).
Lewy, H., A priori limitations for solutions of Monge-Ampère equations. I,Transactions of the American Mathematical Society, Vol. 37, pp. 417–434 (1935).
Lewy, H., A priori limitations for solutions of Monge-Ampère equations, II,Transactions of the American Mathematical Society, Vol. 41, pp. 365–374 (1937).
Lewy, H., On the existence of a closed convex surface realizing a given Riemannian metric,Proceedings of the National Academy of Sciences U.S.A., Vol. 24, No. 2, pp. 104–106 (1938).
Lewy, H., On differential geometry in the large. I (Minkowski's problem),Transactions of the American Mathematical Society, Vol. 43, No. 2, pp. 258–270 (1938).
Lichtenstein, L., Zur Theorie der konformen Abbildung,Bulletin de l'Académie des Sciences de Cracovie, pp. 192–217 (1916).
Miranda, C., Su un problema di Minkowski,Rendiconti del Seminario Matematico di Roma 3, pp. 96–108 (1939).
Miranda, C., Equazioni Alle Derivate Parziali di Tipo Ellittico, Ergebnisse der Mathematik und ihrer Grenzgebiete, Berlin, Springer-Verlag 1955.
Morrey, C. B., On the solutions of quasi-linear elliptic partial differential equations,Transactions of the American Mathematical Society 43, pp. 126–166 (1938).
Nitenberg, L., On non-linear elliptic partial differential equations and Hölder continuity,Communications on Pure and Applied Mathematics, Vol. 6, No. 1, pp. 103–156 (1953).
Nirenberg, L., The Weyl and Minkowski problems in Differential Geometry in the Large,Communications on Pure and Applied Math., Vol. VI, pp. 337–394 (1953).
Pogorelow, A. V., On convex surfaces with regular metric,Doklady Akad. Nauk. S.S.S.R. (N.S.) 67, pp. 791–794 (1949), American Mathematical Society Translation Number 43.
Pogorelow, A. V., Deformation of convex surfaces, Moscow-Leningrad, 1951 (in Russian).
Pogorelow, A. V., Regularity of a convex surface with given Gaussian curvature,Math. Sbornik N.S. 31 (73), pp. 88–103 (1952) (in Russian).
Schauder, J., Ueber lineare elliptische Differentialgleichungen zweiter Ordnung,Mathematische Zeitschrift, Vol. 38, pp. 257–282 (1934).
Schiffer, M. and Spencer, D. C., Functionals of finite Riemann surfaces, Princeton University Press, 1954.
Weyl, H., Ueber die Bestimmung einer geschlossenen konvexen Fläche durch ihr Linienelement,Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich 61, pp. 40–72 (1916), reprinted in Selecta Hermann Weyl, Basel und Stuttgart, 1956, pp. 148–178.
Wintner, A., On Weyl's embedding problem,Proceedings of the National Academy of Sciences of the U.S.A., Vol. 42, No. 3, pp. 157–160 (1956).
Wintner, A., On Weyl's identity in the differential geometry of surfaces,Annali di Matematica Pura ed Applicata, Serie IV, Tomo, XLI, pp. 257–268 (1956).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Heinz, E. On elliptic Monge-Ampère Equations and Weyl’s embedding problem. J. Anal. Math. 7, 1–52 (1959). https://doi.org/10.1007/BF02787679
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02787679