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

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.

The Dirichlet principle for the complex $k$-Hessian functional

“…for u f the solution to (1.5). It follows from [27,30,28,29,36] that Q k is well-defined; it should be regarded as a fully nonlinear analogue of the Dirichlet-to-Neumann map f → (u f ) ν . In terms of this operator, Theorem 1.1 states that (1.12)…”

“…And it is generally believed that Krylov's method could also prove the same results for degenerate complex k-Hessian equations, though we could not find a good reference. (See also [36]). There are many important results on complex Hessian equations on C n and Kähler manifolds, e.g.…”