Abstract

The Lp-Minkowski problem, a generalization of the classical Minkowski problem, was defined by Lutwak in the ’90s. For a fixed real number p ≠ n, it asks what are the necessary and sufficient conditions on a finite Borel measure on Sn−1so that it is the Lp surface area measure of a convex body in Rn. For p = 1, one has the classical Minkowski problem in which the Lp surface area is the usual surface area of a compact set embedded in Rn. Under certain technical assumptions, the planar Lp-Minkowski problem reduces to the study of positive, π-periodic solutions, h : [0, 2π] → (0, ∞) to the non-linear equation h1−p(h′′ + h) = ψ for a given smooth, π-periodic function ψ : [0, 2π] → (0, ∞). In this thesis, we give a new proof of the existence of solutions of the planar Lp-Minkowski problem for 0 < p < 1. To do so, we consider a parabolic anisotropic curvature flow on the space of strictly convex bodies K in R2 which are symmetric with respect to the origin. The case 0 < p < 1 has been considered before by K.S. Chou and X.J. Wang, [5], by studying the corresponding̀̀̀̀̀ Monge-Amprère type equation. The connection between solutions to a parabolic equation, the flow, and a corresponding elliptic equation, the Lp-Minkowski problem, has been long conjectured by the specialists and this is yet another instance where it has been used.