Relaxation in BV under non-standard growth conditions

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.