On second order multiple integrals in the calculus of variations

In this talk we consider variational problems for multiple integrals with integrands containing second order partial derivatives , with fourth order Euler-Lagrange equations. A particular example is the affine area functional , which is invariant under uni-modular affine transformations of the ambient space. We shall discuss the existence and uniqueness of maximisers in appropriate weak formulations, the solvability of boundary value problems, issues of regularity and Bernstein-Liouville type results.