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 Mat. Sb., 1989, Volume 180, Number 7, Pages 867–887 (Mi msb1639)

Solution of the Dirichlet problem for curvature equations of order $m$

N. M. Ivochkina

Abstract: Solvability conditions for curvature equations of order which are sufficient, and almost necessary, are obtained, and theorems concerning the existence of solutions in $C^{l+2+\alpha}(\overline\Omega)$, $l\geqslant2$, $0<\alpha<1$, are proved. The first-order curvature equation coincides with the curvature equation of order $m$, and the curvature equation of order $n$ with the Monge–Ampère equation.
Bibliography: 18 titles.

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English version:
Mathematics of the USSR-Sbornik, 1990, 67:2, 317–339

Bibliographic databases:

UDC: 517.9
MSC: Primary 35J65; Secondary 53C45

Citation: N. M. Ivochkina, “Solution of the Dirichlet problem for curvature equations of order $m$”, Mat. Sb., 180:7 (1989), 867–887; Math. USSR-Sb., 67:2 (1990), 317–339

Citation in format AMSBIB
\Bibitem{Ivo89} \by N.~M.~Ivochkina \paper Solution of the Dirichlet problem for curvature equations of order~$m$ \jour Mat. Sb. \yr 1989 \vol 180 \issue 7 \pages 867--887 \mathnet{http://mi.mathnet.ru/msb1639} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1014618} \zmath{https://zbmath.org/?q=an:0695.35074|0709.35046} \transl \jour Math. USSR-Sb. \yr 1990 \vol 67 \issue 2 \pages 317--339 \crossref{https://doi.org/10.1070/SM1990v067n02ABEH002089} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1990EN23400001} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Trudinger N., “A-Priori Bounds and Necessary Conditions for Solvability of Prescribed Curvature Equations”, Manuscr. Math., 67:1 (1990), 99–112
2. Mi Lin, Neil S. Trudinger, “On some inequalities for elementary symmetric functions”, BAZ, 50:2 (1994), 317
3. Ivochkina N., Ladyzhenskaya O., “The First Initial-Boundary Value-Problem for Evolutionary Equations Generated by Symmetrical Functions of the Principal Curvatures”, Dokl. Akad. Nauk, 340:2 (1995), 155–157
4. Ivochkina N., Tomi F., “Locally Convex Hypersurfaces of Prescribed Curvature and Boundary”, Calc. Var. Partial Differ. Equ., 7:4 (1998), 293–314
5. Bo Guan, Joel Spruck, “Locally convex hypersurfaces of constant curvature with boundary”, Comm Pure Appl Math, 57:10 (2004), 1311
6. Sheng W., Urbas J., Wang X., “Interior Curvature Bounds for a Class of Curvature Equations”, Duke Math. J., 123:2 (2004), 235–264
7. Eric T. Sawyer, Richard L. Wheeden, “Regularity of Degenerate Monge–Ampère and Prescribed Gaussian Curvature Equations in Two Dimensions”, Potential Anal, 24:3 (2006), 267
8. Pierre Bayard, Philippe Delanoë, “Entire spacelike radial graphs in the Minkowski space, asymptotic to the light-cone, with prescribed scalar curvature”, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 26:3 (2009), 903
9. V. N. Kokarev, “Smeshannye formy ob'ema i kompleksnoe uravnenie tipa Monzha–Ampera na tore”, Vestn. SamGU. Estestvennonauchn. ser., 2009, no. 8(74), 35–43
10. V. N. Kokarev, “Mixed volume forms and a complex equation of Monge–Ampère type on Kähler manifolds of positive curvature”, Izv. Math., 74:3 (2010), 501–514
11. N. M. Ivochkina, S. I. Prokof’eva, G. V. Yakunina, “The Gårding cones in the modern theory of fully nonlinear second order differential equations”, J Math Sci, 184:3 (2012), 295
12. Yong Huang, Lu Xu, “Two problems related to prescribed curvature measures”, DCDS-A, 33:5 (2012), 1975
13. N. M. Ivochkina, “From Gårding's cones to $p$-convex hypersurfaces”, Journal of Mathematical Sciences, 201:5 (2014), 634–644
14. Enache C., “Maximum and Minimum Principles for a Class of Monge-Ampere Equations in the Plane, with Applications to Surfaces of Constant Gauss Curvature”, Commun. Pure Appl. Anal, 13:3 (2014), 1347–1359
15. Covei D.-P., “a Necessary and a Sufficient Condition For the Existence of the Positive Radial Solutions To Hessian Equations and Systems With Weights”, Acta Math. Sci., 37:1 (2017), 47–57
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