*Published Paper*

**Inserted:** 4 aug 2014

**Last Updated:** 16 jan 2015

**Journal:** J. Fixed Point Theory Appl.

**Volume:** 15

**Pages:** 537--575

**Year:** 2014

**Notes:**

appears in Haim Brezis festschrift

**Abstract:**

The gradient of any local minimizer of functionals of the type \[ w \to \int_{\Omega} f(x,w,Dw)\, dx+\int_{\Omega} w\mu\, dx, \] 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.

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