Regularity of solutions to the quaternionic Monge-Ampère equation (Q2187696)

The authors study the regularity of solutions to the Dirichlet problem for the quaternionic Monge-Ampère equation. For a domain \(\Omega\subset\mathbb H^n\) let \(\mathcal{QPSH}(\Omega)\) stand for the set of all quaternionic plurisubharmonic functions in \(\Omega\). For \(p>2\) let \(p'\) be the conjugate of \(p\) and let \(\gamma_r:=\frac{r}{r+np'+p'np/(p-2)}\), \(r\geq1\). The main result of the paper is the following regularity theorem. Let \(\Omega\subset\mathbb H^n\) be a quaternionic strictly pseudoconvex domain. Let \(p>2\), \(0\leq f\in L^p(\Omega)\), \(\varphi\in\mathcal C(\partial\Omega)\), and let \(u\in\mathcal{QPSH}(\Omega)\cap\mathcal C(\overline{\Omega})\) be the solution of the Dirichlet problem \((\partial\partial_Ju)^n=f\Omega\), \(u=\varphi\) on \(\partial\Omega\), such that \(\Delta u(\Omega)\) is finite. Assume that there exist \(0<\nu<1\) and \(b\in\operatorname{Lip}_\nu(\overline{\Omega})\) such that \(b\leq u\) in \(\Omega\) and \(b=\varphi\) on \(\partial\Omega\). Then \(u\in\operatorname{Lip}_\alpha(\overline{\Omega})\) for every \(0\leq\alpha<\min\{\nu,2\gamma_1\}\). The proof is based on the following stability theorem for the weak solutions of the quaternionic Monge-Ampère equation. Let \(c_0>0\), \(p>2\), \(u,v\in\mathcal{QPSH}(\Omega)\cap L^\infty_{\operatorname{loc}}(\Omega)\) be such that \(\liminf_{q\to q_0}(u-v)(q)\geq0\) for all \(q_0\in\partial\Omega\) and \(\|(\partial\partial_Ju)^n\|_{L^p(\Omega)}\leq c_0\). Then for any \(r\geq1\) and \(0<\gamma<\gamma_r\) there exists a constant \(C=C(c_0,\gamma,\operatorname{diam}(\Omega))\) such that \(\sup_\Omega(v-u)\leq C\|(v-u)_+\|_{L^r(\Omega)}^\gamma\).