Representation and regularity for the Dirichlet problem for real and complex degenerate Hessian equations

2013/11/25 by Zhou, Wei
#32W50 #35B65 #35J60 #35J70 #Analysis of PDEs (math.AP) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1311.6450

Abstract

We consider the Dirichlet problem for positively homogeneous, degenerate elliptic, concave (or convex) Hessian equations. Under natural and necessary conditions on the geometry of the domain, with the C^1,1 boundary data, we establish the interior C^1,1-regularity of the unique (admissible) solution, which is optimal even if the boundary data is smooth. Both real and complex cases are studied by the unified (Bellman equation) approach.

Related

• The exterior Dirichlet problem for the homogeneous k-Hessian equation
• Hessian equations with elementary symmetric functions
• C^{1, 1} Solution of the Dirichlet Problem for Degenerate k-Hessian Equations
• Hessian continuity at degenerate points in nonvariational elliptic problems
• Second order estimates for convex solutions of degenerate k-Hessian equations