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Mat. Sb. (N.S.), 1985, Volume 128(170), Number 3(11), Pages 403–415 (Mi msb2167)  

This article is cited in 24 scientific papers (total in 24 papers)

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

N. M. Ivochkina


Abstract: The solvability of the problem
$$ F_m(u)=f(x,u,u_x)\geqslant\nu>0,\qquad u|_{\partial\Omega}=0, $$
in $C^{l+2+\alpha}(\overline\Omega)$, $l\geqslant2$, is proved, where $F_m(u)$ is the sum of all the principal minors of order $m$ of the Hessian $F_n(u)\equiv\det(u_{xx})$, $\Omega$ 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.

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English version:
Mathematics of the USSR-Sbornik, 1987, 56:2, 403–415

Bibliographic databases:

UDC: 517.9
MSC: 35Q99
Received: 01.08.1984

Citation: N. M. Ivochkina, “Solution of the Dirichlet problem for some equations of Monge–Aampére type”, Mat. Sb. (N.S.), 128(170):3(11) (1985), 403–415; 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 Mat. Sb. (N.S.)
\yr 1985
\vol 128(170)
\issue 3(11)
\pages 403--415
\mathnet{http://mi.mathnet.ru/msb2167}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=815272}
\zmath{https://zbmath.org/?q=an:0609.35042}
\transl
\jour Math. USSR-Sb.
\yr 1987
\vol 56
\issue 2
\pages 403--415
\crossref{https://doi.org/10.1070/SM1987v056n02ABEH003043}


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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. Ivochkina N., “Solution of the Dirichlet Problem for an Equation of M-Order Curvature”, 299, no. 1, 1988, 35–38  mathscinet  zmath  isi
    2. N. M. Ivochkina, “Solution of the Dirichlet problem for curvature equations of order $m$”, Math. USSR-Sb., 67:2 (1990), 317–339  mathnet  crossref  mathscinet  zmath  isi
    3. Trudinger N., Wang X., “Hessian Measures II”, Ann. Math., 150:2 (1999), 579–604  crossref  mathscinet  zmath  isi
    4. Chou K., Wang X., “A Variational Theory of the Hessian Equation”, Commun. Pure Appl. Math., 54:9 (2001), 1029–1064  crossref  mathscinet  zmath  isi
    5. Hartenstine D., Schmitt K., “On Generalized and Viscosity Solutions of Nonlinear Elliptic Equations”, Adv. Nonlinear Stud., 4:3 (2004), 289–306  mathscinet  zmath  isi
    6. Ivochkina N., Trudinger N., Wang X., “The Dirichlet Problem for Degenerate Hessian Equations”, Commun. Partial Differ. Equ., 29:1-2 (2004), 219–235  crossref  mathscinet  zmath  isi
    7. Chaudhuri N., Trudinger E., “An Alexsandrov Type Theorem Kappa-Convex Functions”, Bull. Aust. Math. Soc., 71:2 (2005), 305–314  crossref  mathscinet  zmath  isi
    8. Wang X., “A Priori Estimates and Existence for a Class of Fully Nonlinear Elliptic Equations in Conformal Geometry”, Chin. Ann. Math. Ser. B, 27:2 (2006), 169–178  crossref  mathscinet  zmath  isi
    9. Sheng W.-M., Trudinger N.S., Wang X.-J., “The Yamabe Problem for Higher Order Curvatures”, J. Differ. Geom., 77:3 (2007), 515–553  mathscinet  zmath  isi
    10. Changyu Ren, “The first initial-boundary value problem for fully nonlinear parabolic equations generated by functions of the eigenvalues of the Hessian”, Journal of Mathematical Analysis and Applications, 339:2 (2008), 1362  crossref
    11. Guan B., “Complete Conformal Metrics of Negative Ricci Curvature on Compact Manifolds with Boundary”, Int. Math. Res. Notices, 2008, rnn105  crossref  mathscinet  zmath  isi
    12. Phuc N.C., Verbitsky I.E., “Singular Quasilinear and Hessian Equations and Inequalities”, J. Funct. Anal., 256:6 (2009), 1875–1906  crossref  mathscinet  zmath  isi  elib
    13. Wang X.-J., “The K-Hessian Equation”, Geometric Analysis and Pdes, Lecture Notes in Mathematics, 1977, eds. Ambrosetti A., Chang S., Malchiodi A., Springer-Verlag Berlin, 2009, 177–252  crossref  mathscinet  zmath  isi
    14. Tian G.-J., Wang X.-J., “Moser-Trudinger Type Inequalities for the Hessian Equation”, J. Funct. Anal., 259:8 (2010), 1974–2002  crossref  mathscinet  zmath  isi
    15. Hou Z., Ma X.-N., Wu D., “A Second Order Estimate for Complex Hessian Equations on a Compact Kahler Manifold”, Math. Res. Lett., 17:3 (2010), 547–561  mathscinet  zmath  isi
    16. N. V. Filimonenkova, “On the classical solvability of the Dirichlet problem for nondegenerate m-Hessian equations”, J Math Sci, 2011  crossref
    17. Rigoli M., Salvatori M., Vignati M., “K-Hessian Differential Inequalities and the Compact Support Principle”, Harmonic Maps and Differential Geometry, Contemporary Mathematics, 542, eds. Loubeau E., Montaldo S., Amer Mathematical Soc, 2011, 151–157  crossref  mathscinet  zmath  isi
    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
    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
    21. 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
    22. 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
    23. 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  mathscinet  isi
    24. 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
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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