Prescribing Capacitary Curvature Measures on Planar Convex Domains

  • Jie XiaoEmail author


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 \(C^{k+2,\alpha }\).

Mathematics Subject Classification

53C45 53C42 52B60 35Q35 31B15 



The author is grateful to Han Hong and Ning Zhang for several discussions on the only-if part of Theorem 1.1(i).


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Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMemorial UniversitySt. John’sCanada

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