On the planar Lp-Minkowski problem.

In this paper, we study the planar L p -Minkowski problem (0.1) u θ θ + u = f u p − 1 , θ ∈ S 1 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 ∈ C α (S 1) , α ∈ (0 , 1) such that (0.1) admits a nonnegative solution vanishes somewhere on S 1. 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 ∈ C α (S 1). 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 ∈ C α (S 1) 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. [ABSTRACT FROM AUTHOR]