  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.
