Quaternionic Monge–Ampère equation

Definition

On a domain $\Omega\subseteq\mathbb H^n$ , a quaternionic Monge–Ampère equation prescribes the quaternionic Hessian measure of an unknown quaternionic plurisubharmonic function $u$ :

\det_M\!\left(\frac{\partial^2u} {\partial\bar q_i\,\partial q_j}\right)=f.

For a nonsmooth $u$ , this equality is interpreted as equality of quaternionic Monge–Ampère measures.

Elliptic branch

The equation is elliptic on the cone where the quaternionic Hessian is positive semidefinite. Requiring $u$ to be quaternionic PSH selects this branch, just as convexity does for the real equation and complex plurisubharmonicity does for the complex equation.

Dirichlet problem

Given boundary data $\varphi$ , the Dirichlet problem asks for

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

Existence and uniqueness hold in the continuous category on bounded strictly quaternionically pseudoconvex domains under the standard nonnegativity hypothesis on $f$ .

References

Semyon Alesker, “Quaternionic Monge–Ampère equations,” Journal of Geometric Analysis 13 (2003), 205–238. arXiv record.
Semyon Alesker, “Quaternionic plurisubharmonic functions and their applications to convexity,” 2016 revision. arXiv record. Relevant: §5.