Viscosity solutions to quaternionic Monge-Ampère equations

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• 作者：Dongrui Wan^a ; ^{wandongrui@szu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Wei Wang^b ; ^{wwang@zju.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Viscosity solutions ; Quaternionic Monge&ndash ; Ampè ; re equations ; Comparison principle
• 刊名：Nonlinear Analysis
• 出版年：2016
• 出版时间：July 2016
• 年：2016
• 卷：140
• 期：Complete
• 页码：69-81
• 全文大小：612 K

文摘

Quaternionic Monge–Ampère equations have recently been studied intensively using methods from pluripotential theory. We present an alternative approach by using the viscosity methods. We study the viscosity solutions to the Dirichlet problem for quaternionic Monge–Ampère equations det(f)=F(q,f)$det\left(f\right)=F(q,f)$ with boundary value f=g$f=g$ on ∂Ω$\partial \Omega$. Here Ω$\Omega$ is a bounded domain on the quaternionic space Hⁿ${\mathbb{H}}^n$, g∈C(∂Ω)$g\in C\left(\partial \Omega \right)$, and F(q,t)$F(q,t)$ is a continuous function on Ω×R→R⁺$\Omega \times \mathbb{R}\to {\mathbb{R}}^{+}$ which is non-decreasing in the second variable. We prove a viscosity comparison principle and a solvability theorem. Moreover, the equivalence between viscosity and pluripotential solutions is shown.