Abstract:
The solvability of the problem
Fm(u)=f(x,u,ux)⩾ν>0,u|∂Ω=0,
in Cl+2+α(¯Ω), l⩾2, is proved, where Fm(u) is the sum of all the principal minors of order m of the Hessian Fn(u)≡det(uxx), Ω is a bounded strictly convex region in Rn, n≥2, with boundary ∂Ω of class Cl+2+α, 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.
\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:
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
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)
Zedong Yang, Zhanbing Bai, “Multiplicity of k-convex solutions for a singular k-Hessian system”, Demonstratio Mathematica, 57:1 (2024)
Chuanqiang Chen, Li Chen, Xinqun Mei, Ni Xiang, “The Neumann problem for a class of mixed complex Hessian equations”, DCDS, 42:9 (2022), 4203
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
Kazuhiro Takimoto, “Precise blowup rate near the boundary of boundary blowup solutions to k-Hessian equation”, SN Partial Differ. Equ. Appl., 2:1 (2021)
Kazuhiro Takimoto, “Second order boundary estimate of boundary blowup solutions to k-Hessian equation”, Journal of Mathematical Analysis and Applications, 500:2 (2021), 125155
Dragos-Patru Covei, “The Keller–Osserman Problem for the k-Hessian Operator”, Results Math, 75:2 (2020)
Xinan Ma, Guohuan Qiu, “The Neumann Problem for Hessian Equations”, Commun. Math. Phys., 366:1 (2019), 1
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
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
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
Saori Nakamori, Kazuhiro Takimoto, Springer Proceedings in Mathematics & Statistics, 176, Geometric Properties for Parabolic and Elliptic PDE's, 2016, 173
Bo Guan, “Second-order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds”, Duke Math. J., 163:8 (2014)
Ivochkina N. Filimonenkova N., “On the Backgrounds of the Theory of M-Hessian Equations”, Commun. Pure Appl. Anal, 12:4 (2013), 1687–1703
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
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
A. Sadullaev, B. Abdullaev, “Potential theory in the class of $m$-subharmonic functions”, Proc. Steklov Inst. Math., 279 (2012), 155–180
N. M. Ivochkina, “From Gårding's cones to $p$-convex hypersurfaces”, Journal of Mathematical Sciences, 201:5 (2014), 634–644
Citation in format AMSBIB
Contact us: