Abstract
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 that the optimal Lorentz space condition \({\mu \in L(n, 1)}\) is satisfied and \({x \rightarrow f(x, \cdot)}\) is suitably Dini continuous.
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To Haïm Brezis, a master of nonlinear analysis
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Baroni, P., Kuusi, T. & Mingione, G. Borderline gradient continuity of minima. J. Fixed Point Theory Appl. 15, 537–575 (2014). https://doi.org/10.1007/s11784-014-0188-x
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DOI: https://doi.org/10.1007/s11784-014-0188-x