Vol. 12, No. 2, 2019

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A viscosity approach to the Dirichlet problem for degenerate complex Hessian-type equations

Sławomir Dinew, Hoang-Son Do and Tat Dat Tô

Vol. 12 (2019), No. 2, 505–535
DOI: 10.2140/apde.2019.12.505
Abstract

A viscosity approach is introduced for the Dirichlet problem associated to complex Hessian-type equations on domains in ${ℂ}^{n}$. The arguments are modeled on the theory of viscosity solutions for real Hessian-type equations developed by Trudinger (1990). As a consequence we solve the Dirichlet problem for the Hessian quotient and special Lagrangian equations. We also establish basic regularity results for the solutions.

Keywords
nonlinear elliptic PDEs, viscosity solutions, pluripotential theory
Primary: 35J70
Secondary: 32U15
Milestones
Accepted: 30 May 2018
Published: 14 August 2018
Authors
 Sławomir Dinew Institute of Mathematics Jagiellonian University Kraków Poland Hoang-Son Do Institute of Mathematics Vietnam Academy of Science and Technology Hanoi Vietnam Tat Dat Tô Institut Mathématiques de Toulouse Université Paul Sabatier Toulouse France