On $L^\infty$ estimates for fully nonlinear partial differential equations

Abstract

Sharp L ∞ estimates are obtained for general classes of fully non-linear PDE's on Hermitian manifolds. The method builds on the method of comparison with auxiliary Monge-Ampère equations introduced earlier in the Kähler case by the authors in joint work with F. Tong, but uses this time auxiliary equations on open balls with a Dirichlet condition. The role of Yau's theorem in the compact Kähler case is now played by the theorem of Caffarelli-Kohn-Nirenberg-Spruck for the Dirichlet problem. Global estimates are then deduced by combining arguments of Blocki with exponential estimates due to Kolodziej generalizing the classic inequality of Brezis-Merle. The method applies not just to compact Hermitian manifolds, but also to the Dirichlet problem, to open manifolds with a lower bound on their injectivity radii, and to (n − 1)-form Monge-Ampère equations.

Diameter estimates in Kähler geometry

“…[10,16,22,25,31]). However, there has been considerable progress recently in PDE methods for 𝐿 ∞ estimates for fully nonlinear equations [11,12,14]. These new methods turn out to be particularly amenable to geometric estimates, and have been shown to imply some promising estimates for noncollapse [17] and for the Green's function [13].…”

“…This requires an argument by contradiction (Section 5), which is quite different from those in Refs. [11,13,14], and which may be of independent interest.…”

“…In fact, by the Hölder-Young inequality (see, e.g., Equation (2.24) in Ref. [11]), there exists 𝐶 > 0 such that for any 𝛿 > 0 and smooth functions 𝑢 and 𝐹, we have…”

Uniform entropy and energy bounds for fully non-linear equations

“…The proof of the L ∞ bound for fully non-linear PDE's which we just described was based on a comparison with an auxiliary Monge-Ampère equation involving integrals A s on the sublevel sets Ω s = {z ∈ X; ϕ < −s} of the function ϕ. This method turns out to be particularly effective: it has been extended by various authors to stability estimates [12], nef classes [13], moduli of continuity [14,18], lower bounds for the Green's function [15,17], as well as Hermitian manifolds [16] and parabolic equations [5].…”

Sharp uniform bound for the quaternionic Monge-Ampère equation on hyperhermitian manifolds

“…The main goal of the present paper is to derive such a sharp uniform estimates for the quaternionic Monge-Ampère and certain other quaternionic equations. For this, we apply the method of Guo and Phong [10] for fully non-linear equations satisfying a specific structural condition on Hermitian manifolds. In our case, the role of this structural condition is played by a new inequality between complex and quaternionic Monge-Ampère operators, which can be viewed as the quaternionic version of the classic inequality of Cheng and Yau, cf.…”