Relaxation in BV under non-standard growth conditions
Abstract
Morrey's lower semicontinuity theorem for quasiconvex integrands is a
classical result that establishes the existence of minimisers to
variational problems by the Direct Method, provided the integrand
satisfies "standard" growth conditions (i.e. when the growth and
coercivity exponents match). This theorem has more recently been refined
to consider convergence in Sobolev Spaces below the growth exponent of
the integrand: such results can be used to show existence of solutions
to a "Relaxed minimisation problem" when we have "non-standard'" growth
conditions.
When the integrand satisfies linear coercivity
conditions, it is much more useful to consider the space of functions of
Bounded Variation, which has better compactness properties than
$W^{1,1}$. We review the key results in the standard growth case, before
giving an overview of recent results that we have obtained in the
non-standard case. We find that new techniques and ideas are required in
this setting, which in fact provide us with some interesting (and
perhaps unexpected) corollaries on the general nature of quasiconvex
functions.