Borderline gradient continuity of minima
  The gradient of any local minimiser of functionals of the type $$ w \mapsto \int_\Omega f(x,w,Dw)\,dx+\int_\Omega w\mu\,dx, $$ where $f$ has $p$-growth, $p>1$, and $\Omega \subset \mathbb R^n$, is continuous provided the optimal Lorentz space condition $\mu \in L(n,1)$ is satisfied and $x\to f(x, \cdot)$ is suitably Dini-continuous. 
Paolo Baroni, Tuomo Kuusi, Giuseppe Mingione
math.AP