Regularity theory in the calculus of variations and elliptic systems - Irving, C (2022)
In this thesis we will investigate the regularity of critical points and minimisers of integral functionals of the form \[ \mathcal{F}(w) = F (x, w, \nabla w) dx, \] where the mappings w we consider will be vector-valued. This will be done from several different perspectives, with the goal of understanding when solutions are locally regular (of class C^{1,α} ) near a given point $x_0 \in \Omega$. This is typically done through a so-called “ε-regularity result,” which characterises the points x_0 which are regular in the above sense, by a smallness condition for a suitable excess quantity. We will investigate in three different aspects such regularity issues.