On $L^\infty$ estimates for fully nonlinear partial differential equations
Abstract
Sharp estimates are obtained for general classes of fully non-linear PDE's on non-K\"ahler manifolds, complementing the theory developed earlier by the authors in joint work with F. Tong for the K\"ahler case. The key idea is still a comparison with an auxiliary Monge-Amp\`ere equation, but this time on a ball with Dirichlet boundary conditions, so that it always admits a unique solution. The method applies not just to compact Hermitian manifolds, but also to the Dirichlet problem, to open manifolds with a positive lower bound on their injectivity radii, to form equations, and even to non-integrable almost-complex or symplectic manifolds. It is the first method applicable in any generality to large classes of non-linear equations, and it usually improves on other methods when they happen to be available for specific equations.
Cite
@article{arxiv.2204.12549,
title = {On $L^\infty$ estimates for fully nonlinear partial differential equations},
author = {Bin Guo and Duong H. Phong},
journal= {arXiv preprint arXiv:2204.12549},
year = {2023}
}
Comments
This is an expanded version, which also treats gradient terms in $(n-1)$-form equations, and equations on almost-complex and almost-K\"ahler manifolds