Methods and Applications of Analysis

Volume 27 (2020)

Number 4

Special Issue for the 60th Birthday of John Urbas: Part I

Guest Editors: Neil Trudinger and Xu-Jia Wang

Optimal transport and the Gauss curvature equation

Pages: 387 – 404

DOI: https://dx.doi.org/10.4310/MAA.2020.v27.n4.a5

Authors

Nestor Guillen (Department of Mathematics, Texas State University, San Marcos, Tx., U.S.A.)

Jun Kitagawa (Department of Mathematics, Michigan State University, East Lansing, Mich., U.S.A.)

Abstract

In this short note, we consider the problem of prescribing the Gauss curvature and image of the Gauss map for the graph of a function over a domain in Euclidean space. The prescription of the image of the Gauss map turns this into a second boundary value problem. Our main observation is that this problem can be posed as an optimal transport problem where the target is a subset of the lower hemisphere of $\mathbb{S}^n$. As a result we obtain existence and regularity of solutions under mild assumptions on the curvature, as well as a quantitative version of a gradient blowup result due to Urbas, which turns out to fall within the optimal transport framework.

Keywords

Gauss curvature, convex surfaces, optimal transport, Monge–Ampére equations

2010 Mathematics Subject Classification

35J96, 49Qxx, 53C42, 53C45

NG’s research was supported in part by National Science Foundation Grant DMS-1700307.

JK’s research was supported in part by National Science Foundation grants DMS-1700094 and DMS-2000128.

Received 14 May 2020

Accepted 29 October 2020

Published 24 September 2021