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

Bin Guo; Duong H. Phong

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arXiv · 2204.12549

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

Bin Guo·Duong H. Phong

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Abstract

Sharp $L^\infty$ 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 $(n-1)$ 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.

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Bin Guo, Duong H. Phong. 2022-04-26. On $L^\infty$ estimates for fully nonlinear partial differential equations. https://arxiv.org/abs/2204.12549

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