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Mathematics of the USSR-Sbornik, 1987, Volume 56, Issue 2, Pages 403–415
DOI: https://doi.org/10.1070/SM1987v056n02ABEH003043
(Mi sm2167)
 

This article is cited in 38 scientific papers (total in 38 papers)

Solution of the Dirichlet problem for some equations of Monge–Aampére type

N. M. Ivochkina
References:
Abstract: The solvability of the problem
Fm(u)=f(x,u,ux)ν>0,u|Ω=0,
in Cl+2+α(¯Ω), l2, is proved, where Fm(u) is the sum of all the principal minors of order m of the Hessian Fn(u)det, \Omega is a bounded strictly convex region in R^n, n\geq2, with boundary \partial\Omega of class C^{l+2+\alpha}, for m = 1,2,3,n, under certain restrictions on the occurrence of u and p as arguments in f(x,u,p).
Bibliography: 21 titles.
Received: 01.08.1984
Bibliographic databases:
UDC: 517.9
MSC: 35Q99
Language: English
Original paper language: Russian
Citation: N. M. Ivochkina, “Solution of the Dirichlet problem for some equations of Monge–Aampére type”, Math. USSR-Sb., 56:2 (1987), 403–415
Citation in format AMSBIB
\Bibitem{Ivo85}
\by N.~M.~Ivochkina
\paper Solution of the Dirichlet problem for some equations of Monge--Aamp\'ere type
\jour Math. USSR-Sb.
\yr 1987
\vol 56
\issue 2
\pages 403--415
\mathnet{http://mi.mathnet.ru/eng/sm2167}
\crossref{https://doi.org/10.1070/SM1987v056n02ABEH003043}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=815272}
\zmath{https://zbmath.org/?q=an:0609.35042}
Linking options:
  • https://www.mathnet.ru/eng/sm2167
  • https://doi.org/10.1070/SM1987v056n02ABEH003043
  • https://www.mathnet.ru/eng/sm/v170/i3/p403
  • This publication is cited in the following 38 articles:
    1. Zedong Yang, Zhanbing Bai, “Existence results for the k-Hessian type system with the gradients via R+n-monotone matrices”, Nonlinear Analysis, 240 (2024), 113457  crossref
    2. Xingyue He, Chenghua Gao, Jingjing Wang, “k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities”, Advances in Nonlinear Analysis, 13:1 (2024)  crossref
    3. Zedong Yang, Zhanbing Bai, “Multiplicity of k-convex solutions for a singular k-Hessian system”, Demonstratio Mathematica, 57:1 (2024)  crossref
    4. Chuanqiang Chen, Li Chen, Xinqun Mei, Ni Xiang, “The Neumann problem for a class of mixed complex Hessian equations”, DCDS, 42:9 (2022), 4203  crossref
    5. Kazuhiro Takimoto, “Bernstein type theorem for the generalized parabolic 2-Hessian equation under weaker conditions”, Journal of Mathematical Analysis and Applications, 495:1 (2021), 124703  crossref
    6. Kazuhiro Takimoto, “Precise blowup rate near the boundary of boundary blowup solutions to k-Hessian equation”, SN Partial Differ. Equ. Appl., 2:1 (2021)  crossref
    7. Kazuhiro Takimoto, “Second order boundary estimate of boundary blowup solutions to k-Hessian equation”, Journal of Mathematical Analysis and Applications, 500:2 (2021), 125155  crossref
    8. Dragos-Patru Covei, “The Keller–Osserman Problem for the k-Hessian Operator”, Results Math, 75:2 (2020)  crossref
    9. Xinan Ma, Guohuan Qiu, “The Neumann Problem for Hessian Equations”, Commun. Math. Phys., 366:1 (2019), 1  crossref
    10. 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  crossref  mathscinet  zmath  isi
    11. N. M. Ivochkina, N. V. Filimonenkova, “Konusy Gordinga i uravneniya Bellmana v teorii gessianovskikh operatorov i uravnenii”, Differentsialnye i funktsionalno-differentsialnye uravneniya, SMFN, 63, no. 4, Rossiiskii universitet druzhby narodov, M., 2017, 615–626  mathnet  crossref
    12. Dragos Patru Covei, “A remark on the existence of entire large and bounded solutions to a (k 1, k 2)-Hessian system with gradient term”, Acta. Math. Sin.-English Ser., 33:6 (2017), 761  crossref
    13. Saori Nakamori, Kazuhiro Takimoto, Springer Proceedings in Mathematics & Statistics, 176, Geometric Properties for Parabolic and Elliptic PDE's, 2016, 173  crossref
    14. Saori Nakamori, Kazuhiro Takimoto, “A Bernstein type theorem for parabolic <mml:math altimg="si1.gif" display="inline" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd" xmlns:sa="http://www.elsevier.com/xml/common/struct-aff/dtd"><mml:mi>k</mml:mi></mml:math>-Hessian equations”, Nonlinear Analysis: Theory, Methods & Applications, 117 (2015), 211  crossref  zmath
    15. Bo Guan, “Second-order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds”, Duke Math. J., 163:8 (2014)  crossref
    16. Ivochkina N. Filimonenkova N., “On the Backgrounds of the Theory of M-Hessian Equations”, Commun. Pure Appl. Anal, 12:4 (2013), 1687–1703  crossref  mathscinet  zmath  isi
    17. N. M. Ivochkina, “On some properties of the positive m-Hessian operators in C 2(Ω)”, J. Fixed Point Theory Appl., 14:1 (2013), 79  crossref
    18. 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  crossref  mathscinet  zmath
    19. A. Sadullaev, B. Abdullaev, “Potential theory in the class of m-subharmonic functions”, Proc. Steklov Inst. Math., 279 (2012), 155–180  mathnet  crossref  mathscinet  isi  elib
    20. N. M. Ivochkina, “From Gårding's cones to p-convex hypersurfaces”, Journal of Mathematical Sciences, 201:5 (2014), 634–644  mathnet  crossref  mathscinet
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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