Prescribing capacitary curvature measures on planar convex domains

TL;DR

This paper characterizes when a convex planar domain can be prescribed to have a specific $p$-capacitary curvature measure, establishing existence, uniqueness, and regularity conditions for the solution.

Contribution

It provides necessary and sufficient conditions for prescribing $p$-capacitary curvature measures on convex domains, including regularity results and uniqueness up to translation.

Findings

Prescribing $mbda_p$ measure is solvable iff measure has centroid at origin and no antipodal support points.

Solution is unique up to translation.

Boundary regularity depends on the regularity of the prescribed measure.

Abstract

For $p\in (1,2]$ and a bounded, convex, nonempty, open set $\Omega\subset\mathbb R^2$ let $\mu_p(\bar{\Omega},\cdot)$ be the $p$-capacitary curvature measure (generated by the closure $\bar{\Omega}$ of $\Omega$) on the unit circle $\mathbb S^1$. This paper shows that such a problem of prescribing $\mu_p$ on a planar convex domain: "Given a finite, nonnegative, Borel measure $\mu$ on $\mathbb S^1$, find a bounded, convex, nonempty, open set $\Omega\subset\mathbb R^2$ such that $d\mu_p(\bar{\Omega},\cdot)=d\mu(\cdot)$" is solvable if and only if $\mu$ has centroid at the origin and its support $\mathrm{supp}(\mu)$ does not comprise any pair of antipodal points. And, the solution is unique up to translation. Moreover, if $d\mu_p(\bar{\Omega},\cdot)=\psi(\cdot)\,d\ell(\cdot)$ with $\psi\in C^{k,\alpha}$ and $d\ell$ being the standard arc-length element on $\mathbb S^1$, then $\partial\Omega$ is of $C^{k+2,\alpha}$.