Prescribing capacitary curvature measures on planar convex domains

For $pin (1,2]$ and a bounded, convex, nonempty, open set $Omegasubsetmathbb R^2$ 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 $Omegasubsetmathbb R^2$ such that $dmu_p(bar{Omega},cdot)=dmu(cdot)$ 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 $dmu_p(bar{Omega},cdot)=psi(cdot),dell(cdot)$ with $psiin C^{k,alpha}$ and $dell$ being the standard arc-length element on $mathbb S^1$, then $partialOmega$ is of $C^{k+2,alpha}$.