Prescribing Capacitary Curvature Measures on Planar Convex Domains

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 $C^{k+2,\alpha}$.