Abstract
We present a viscosity approach to the Dirichlet problem for the complex Monge–Ampère equation \({\det u_{\bar{j} k} = f (x, u)}\) . Our approach differs from previous viscosity approaches to this equation in several ways: it is based on contact set techniques (the Alexandrov–Bakelman–Pucci estimate), on extensive applications of sup-inf convolutions, and on a relation between real and complex Hessians. More specifically, this paper includes a notion of viscosity solutions; a comparison principle and a solvability theorem; the equivalence between viscosity and pluripotential solutions; an estimate of the modulus of continuity of a solution in terms of that of a given subsolution and of the right-hand side f; and an Alexandrov–Bakelman–Pucci type of L ∞-estimate.
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Wang, Y. A viscosity approach to the Dirichlet problem for complex Monge–Ampère equations. Math. Z. 272, 497–513 (2012). https://doi.org/10.1007/s00209-011-0946-z
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DOI: https://doi.org/10.1007/s00209-011-0946-z