Rarity of $\boldsymbol{\mathcal{C}^{1,1}}$ solutions to the complex Monge--Amp\`ere equation on weakly pseudoconvex domains

TL;DR

This paper demonstrates that on certain weakly pseudoconvex domains, the classical Dirichlet problem for the complex Monge--Ampère equation with smooth data generally lacks solutions with ,1 regularity, impacting extension problem approaches.

Contribution

It establishes the non-existence of ,1 solutions for the complex Monge--Ampère equation on weakly pseudoconvex B-regular domains, challenging previous assumptions.

Findings

,1 solutions do not generally exist on weakly pseudoconvex B-regular domains.

The result affects potential-theoretic approaches to extension problems for mappings.

The work highlights limitations of classical regularity assumptions in complex Monge--Ampre theory.

Abstract

We show that on any weakly pseudoconvex $B$-regular domain, the classical Dirichlet problem for the complex Monge--Amp\`ere equation with $\mathcal{C}^\infty$-smooth data does not in general admit $\mathcal{C}^{1,1}$-smooth solutions. This working draft is a prelude to potential-theoretic solutions to some extension problems for mappings that were thought to rely on such $\mathcal{C}^{1,1}$-smooth solutions.