The Dirichlet Problem for the Homogenous Mixed Hessian Type Equation in a Punctured Domain

Abstract

In this paper, we investigate the Dirichlet problem concerning the homogeneous mixed Hessian type equation in the convex cone ${\widetilde \Gamma}_{k}$ with prescribed asymptotic behavior at 0 ∈ Ω, where Ω is a strictly (k − 1)-convex bounded domain. By constructing approximating solutions, we establish the corresponding theorems on the existence and uniqueness of C^1,1 solutions. The key is to construct subsolutions for the approximating non-degenerate mixed Hessian type equation. Moreover, our main technique is to establish uniform gradient estimates and second-order estimates which are independent of the approximation.

Keywords

Mixed Hessian type equation / ${\widetilde \Gamma}_{k}$-admissible solution / prescribed asymptotic behavior / a priori estimate / 35J60 / 35B45

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