On the planar L-Minkowski problem

Abstract

In this paper, we study the planar L p -Minkowski problem (0.1)for all p ∈ R, which was introduced by Lutwak [23]. 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 [8]. 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 [8] for p < 2. At the end of this paper, some new uniqueness results will also be shown.

Anisotropic flow, entropy, and -Minkowski problem

“…The p = −n − 1 (or equivalently, α = 1 n+2 ) case of the L p -Minkowski problem is the critical case because its link with the SL(n) invariant centro-affine curvature whose reciprocal is u n+2 det( ∇2 i j u + u ḡi j ) (see [29] or [38]). For positive results concerning the critical case p = −n − 1, see, for example, [28,34], and for obstructions for a solution, see, for example, [20,22].…”

“…In the super-critical case p < −n − 1 (or equivalently, α < 1 n+2 ), there is a recent important work by Li, Guang, and Wang [27] proving that for any positive C 2 function f, there exists a C 4 solution of (1.3). See also [22] for non-existence examples.…”