Open Collections

A bridge between the Sobolev and Sobolev trace inequalities and beyond

Banff International Research Station for Mathematical Innovation and Discovery

In this talk, we show that the classical Sobolev and Sobolev trace inequalities are embedded into the same one-parameter family of sharp constrained Sobolev inequalities on half-spaces. Using a new variation of a mass transportation argument introduced by Cordero-Erausquin, Nazaret, and Villani, we prove each inequality in this one-parameter family and characterize the equality cases. The case $p=2$ corresponds to a family of variational problems on conformally flat metrics, whose absolute minimizers interpolate between conformally flat spherical and hyperbolic geometries, passing through the Euclidean geometry defined by the fundamental solution of the Laplacian. This is joint work with Francesco Maggi.

Mathematics; Partial differential equations; Calculus of variations and optimal control; optimization

video/mp4

Author affiliation: University of Texas

2017-10-11

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/63242

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/