Dirichlet theorem for the quaternionic Monge–Ampère equation

Statement

Let $\Omega\subseteq\mathbb H^n$ be a bounded strictly quaternionically pseudoconvex domain. If $f\in C(\overline\Omega)$ is nonnegative and $\varphi\in C(\partial\Omega)$ , then there is a unique $u\in C(\overline\Omega)$ , quaternionic plurisubharmonic on $\Omega$ , such that

\operatorname{MA}_{\mathbb H}(u)=f\,dV, \qquad u|_{\partial\Omega}=\varphi.

Interpretation

The interior equation is an equality of quaternionic Monge–Ampère measures. Thus the theorem includes degenerate right-hand sides that may vanish, and the solution need not be $C^2$ .

Smooth ball case

When $\Omega$ is the unit ball, $f>0$ and both $f$ and $\varphi$ are smooth, the solution is smooth up to the boundary. This stronger regularity statement should not be silently extended to arbitrary continuous data.

References

Semyon Alesker, “Quaternionic Monge–Ampère equations,” Journal of Geometric Analysis 13 (2003), 205–238. arXiv record. Relevant: Theorems 0.1.2–0.1.3.
Semyon Alesker, “Quaternionic plurisubharmonic functions and their applications to convexity,” 2016 revision. arXiv record. Relevant: Theorems 5.2 and 5.4.