The local Burkholder functional, quasiconvexity and Geometric Function Theory

TL;DR

This paper proves the quasiconvexity of the local Burkholder functional, introduces new non-polyconvex quasiconvex functionals near p=2, and establishes existence of minimizers in geometric function theory.

Contribution

It demonstrates quasiconvexity of the local Burkholder functional and identifies new classes of non-polyconvex quasiconvex functionals with minimizers.

Findings

The local Burkholder functional is quasiconvex.

New non-polyconvex quasiconvex functionals are found near p=2.

Existence of minimizers for certain non-polyconvex functionals.

Abstract

We show that the local Burkholder functional $\mathcal B_K$ is quasiconvex. In the limit of $p$ going to 2 we find a class of non-polyconvex functionals which are quasiconvex on the set of matrices with positive determinant. In order to prove the validity of lower semicontinuity arguments in this setting, we show that the Burkholder functionals satisfy a sharp extension of the classical function theoretic area formula. As a corollary, in addition to functionals in geometric function theory, one finds new classes of non-polyconvex functionals, degenerating as the determinant vanishes, for which there is existence of minimizers.