The Dirichlet problem for Fully Nonlinear Equations Arising from Conformal Geometry

Weisong Dong

The Dirichlet problem for Fully Nonlinear Equations Arising from Conformal Geometry

Analysis of PDEs 2018-02-06 v2

Authors: Weisong Dong

Abstract

We study the Dirichlet problem for a class of curvature equations arising from conformal geometry on Riemannian manifolds $(M^n, g)$ with boundary where $n \geq 3$ . We prove there exists a unique solution using the continuity method which is based on \emph{a priori} estimates for admissible solutions. In deriving the estimates, a crucial step is to derive a lower bound for the gradient on the boundary. This is overcome by constructing a cluster of subsolutions.

Keywords

dirichlet problem boundary value problems partial differential equations

Cite

@article{arxiv.1801.05140,
  title  = {The Dirichlet problem for Fully Nonlinear Equations Arising from Conformal Geometry},
  author = {Weisong Dong},
  journal= {arXiv preprint arXiv:1801.05140},
  year   = {2018}
}

Comments

The main result is similar to Corollary 5.4 in the paper "Complete Conformal Metrics of Negative Ricci Curvature on Compact Manifolds with Boundary" by Bo Guan

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