Regularity and uniqueness of minimizers in the quasiconvex case

In the context of integral functionals defined over a Sobolev
class of the type \(W^{1,p}_g(\Omega,\mathbb{R}^N)\), with
\(N\geq 1\), the quasiconvexity of the integrand is known to be
equivalent to the lower semicontinuity of the functional. In this
context, L.C. Evans showed in 1986 that the minimizers are
regular outside a subset of their domain of measure zero. On the
other hand, E. Spadaro recently provided examples showing that no
uniqueness of minimizers can be expected even under strong
quasiconvexity assumptions. In this talk we will show that, under
the same natural assumptions on the integrand, if the boundary
conditions are suitably small, it is possible to obtain full
regularity (up to the boundary) for the minimizers and,
furthermore, we establish that they are unique. This is joint
work with Jan Kristensen.