A construction of quasiconvex functions

It is well-known that polyconvexity implies quasiconvexity, and that the converse is false. In fact, the class of quasiconvex functions on two by two matrices R^2x2 is in a sense much larger than the corresponding class of polyconvex functions. Despite this there are not many explicit examples of quasiconvex, non-polyconvex functions on R^2x2 in the literature. Besides this there are interesting problems that can be restated as problems about quasiconvexity of associated functions, and it is often easy to show that these functions are rank-one convex, but that they cannot be polyconvex.

In this talk, which is based on joint work with Tadeusz Iwaniec, we present a construction which yields explicit examples of quasiconvex, non-polyconvex functions. The construction is based on the observation that given any suitably rank-one convex function R and any strongly quasiconvex function P the function R+tP is quasiconvex for sufficiently large numbers t. We regard the function R as the function that we ideally would like to show is quasiconvex and regard tP as a perturbation. The method is illustrated on two well-known families of functions, and it involves local polyconvexity.