Prescribing capacitary curvature measures on planar convex domains

Abstract

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

15 pages