On the planar Lp-Minkowski problem

Abstract

In this paper, we study the planar Lp-Minkowski problem(0.1)uθθ+u=fup−1,θ∈S1 for all p∈R, which was introduced by Lutwak [21]. A detailed exploration of (0.1) on solvability will be presented. More precisely, we will prove that for p∈(0,2), there exists a positive function f∈Cα(S1),α∈(0,1) such that (0.1) admits a nonnegative solution vanishes somewhere on S1. In case p∈(−1,0], a surprising a-priori upper/lower bound for solution was established, which implies the existence of positive classical solution to each positive function f∈Cα(S1). When p∈(−2,−1], the existence of some special positive classical solution has already been known using the Blaschke-Santalo inequality [7]. Upon the final case p≤−2, we show that there exist some positive functions f∈Cα(S1) such that (0.1) admits no solution. Our results clarify and improve largely the planar version of Chou-Wang's existence theorem [7] for p<2. At the end of this paper, some new uniqueness results will also be shown.