Workshop on the h-principle and beyond

A topological view on the Monge-Ampere equation without convexity assumptions

Abstract: In this talk we consider the classical Monge-Amp´ere equation in two dimensions in a low-regularity regime:

(0.1) det D 2u = f on D R2 .

We will assume that f is a given strictly positive, smooth function, but we want to assume as little regularity as possible on u. For instance we are interested in the situation u C1, 1/2. In particular we don’t make any assumptions on the convexity of u.

First we will recall shortly how (0.1) can interpreted under these assumptions from a topological point of view. Thereafter we will give an idea how this information can be transformed into a regularity results of u i.e. that it (or u) coincides with the classical Alexandrov solution.

If time permits we will report on its consequence for the rigidity of very weak solutions to the two-dimensional Monge-Amp ´ere equation and two-dimensional isometric embeddings.

The talk is about work in progress.

Date & Time

November 04, 2021 | 10:15am – 11:15am

Location

Simonyi 101 and Remote Access

Speaker Affiliation

University of Leipzig

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