Rank-one convexity: Geometry of matrix space and applications to PDE
Laszlo Szekelyhidi
Institute for Advanced Study
A function f:Rmxn->R is said to be rank-one convex if it is convex
along rank 1 directions. Rank-one convexity gives rise to a variety of
interesting issues such as how to characterize probability measures
that satisfy Jensen's inequality for all rank-one convex functions,
and how to tell when the rank-one convex hull of a given set of
matrices is trivial. These issues in turn can be of great help in
constructing (using convex integration) weak solutions to elliptic
PDEs. In the talk I will present this on two examples: critical points
to polyconvex functionals that are Lipschitz but nowhere C^1, and
solutions to isotropic equations that are exactly at the threshold of
higher integrability.