SECOND-ORDER ESTIMATES AND REGULARITY FOR FULLY NONLINEAR ELLIPTIC EQUATIONS ON RIEMANNIAN MANIFOLDS

By Bo Guan. 2012

190 citations recorded by the source corpus.

We derive a priori second order estimates for solutions of a class of fully nonlinear elliptic equations on Riemannian manifolds under some very general structure conditions. We treat both equations on closed manifolds, and the Dirichlet problem on manifolds with boundary without any geometric restrictions to the boundary except being smooth and compact. As applications of these estimates we obtain results on regularity and existence.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f438a04ffcd18d65125fb41fc68b03020e0042c0

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

A class of curvature type equations. (2019, methodology): “Therefore, the lemmas in the proof only relying on the algebraic properties follow the way same as [19, 20].”
Gauduchon metrics with prescribed volume form (2015, background): “Some other works on this topic include [6, 7, 13, 14, 15, 16, 21, 22, 23, 25, 26, 30, 31, 39, 40, 42, 43, 44].”
Concavity of the Lagrangian phase operator and applications (2016, methodology): “Since our operator does not quite fit the structural conditions imposed in [9, 11] (see”
A second order estimate for general complex Hessian equations (2015, background): “The real counterparts of (1-2) for 1< k ≤ n, with ψ not depending on the gradient of u, have been studied extensively in the literature (see the survey paper [Wang 2009] and more recent related work [Guan 2014]), as these equations appear naturally and play very important roles in both classical and”
Fully non-linear parabolic equations on compact Hermitian manifolds (2017, methodology): “The original strategy goes back to the work of Chou-Wang [6], with adaptation to complex Hessian equations by Hou-Ma-Wu [25], and to fully non-linear elliptic equations admitting a C-subsolution by Guan [19] and Székelyhidi [50].”
Oblique boundary value problems for augmented Hessian equations I (2015, background): “On the other hand, the classical Dirichlet problem for basic Hessian equations has been well studied in the wake of fundamental papers by Caffarelli, Nirenberg and Spruck [1] and Ivochkina [8], with further key developments by several authors, including Krylov in [17] and related papers and Trudinge”
On Bismut flat manifolds (2016, unknown): “We refer the readers to [37], [9], [10], [11], [12], [13], [24], [41], [19], [16], [17], [18], [33], [34], [35], [36], [25], [26], [27], [39], [40], [38] and the references therein for some recent progress in this area.”
The global curvature estimate for the n-2 Hessian equation (2020, background): “The curvature estimate was also established for the equation of the prescribing curvature measure problem by Guan-Li-Li [24] and Guan-Lin-Ma [23].”