Prescribing Capacitary Curvature Measures on Planar Convex Domains

Abstract

For p ∈ (1, 2] and a bounded, convex, nonempty, open set Ω ⊂ R 2 let µ p (Ω, ·) be the p-capacitary curvature measure (generated by the closureΩ of Ω) on the unit circle 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 S 1 , find a bounded, convex, nonempty, open set Ω ⊂ R 2 such that dµ p (Ω, ·) = dµ(·)" is solvable if and only if µ has centroid at the origin and its support supp(µ) does not comprise any pair of antipodal points. And, the solution is unique up to translation. Moreover, if dµ p (Ω, ·) = ψ(·) dℓ(·) with ψ ∈ C k,α and dℓ being the standard arc-length element on S 1 , then ∂Ω is of C k+2,α .