Prescribing harmonic measure on convex domains (Q1177905)

In this very interesting paper the author studies an analog of the classical Minkowski problem in which surface measure is replaced by harmonic measure. More precisely, let \(\Omega\) be a convex, open subset of \(\mathbb{R}^ N\) and \(S^{N-1}\) the surface of the unit ball in \(\mathbb{R}^ N\). Denote by \(d\omega\) the harmonic measure for \(\Omega\) at \(0(\in\Omega)\). Define a measure \(\mu\) on \(S^{N-1}\) by \[ \mu(E)=\omega(g^{-1}(E)\quad\text{for all Borel sets } E\subset S^{N-1}, \] where \(g: \partial\Omega\to S^{N-1}\) is the Gauss map; that is \(g_ *(d\omega)=d\mu\). The problem is: Given a probability measure \(\mu\) on \(S^{N-1}\), find a convex open set \(\Omega\) such that \(g_ *(d\omega)=d\mu\). The main result is existence, uniqueness and regularity in Hölder classes: Theorem: For any non-negative integer \(k\) and any \(\alpha\), \(0<\alpha<1\), let \(R\in C^{k,\alpha}(S^{N-1})\) be positive and satisfy \(\int_{S^{N-1}} Rd\theta=1\), where \(d\theta\) is the standard volume element on \(S^{N-1}\). There exists a bounded, strongly convex open subset \(\Omega\) of \(\mathbb{R}^ N\), containing the origin, with \(C^{k+2,\alpha}\)-boundary, such that for all Borel sets \(E\subset S^{N-1}\), \[ \omega(g^{-1}(E))=\int_ E Rd\theta. \] Moreover, the domain \(\Omega\) is unique up to dilation. The proof makes use of the regularity theory in the Minkowski problem, of recent sharp estimates for the Monge-Ampère equation and on a new doubling property for harmonic measure in convex domains.