Borderline gradient continuity of minima

The gradient of any local minimiser of functionals of the type $w \mapsto \int_Ωf(x,w,Dw)\,dx+\int_Ωwμ\,dx,$ where $f$ has $p$-growth, $p>1$, and $Ω\subset \mathbb R^n$, is continuous provided the optimal Lorentz space condition $μ\in L(n,1)$ is satisfied and $x\to f(x, \cdot)$ is suitably Dini-continuous.

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