GRADIENT REGULARITY FOR NON-AUTONOMOUS FUNCTIONALS WITH DINI OR NON-DINI CONTINUOUS COEFFICIENTS.

We prove C1regularity for local vectorial minimizers of the nonautonomous functional w ∈ W1,1loc (Ω; RN) 7→ Z Ω b(x) |Dw|p + a(x)|Dw|p log(e + |Dw|)dx, with Ω open subset of Rn, n ≥ 2, p > 1, 0 ≤ a(·) ≤ kakL∞(Ω) < ∞, and 0 < ν ≤ b(·) ≤ L. The result is valid provided that the function a(·) is logDini continuous and that the coefficient b(·) is Dini continuous or it is weakly differentiable and its gradient locally belongs to the Lorentz space Ln,1 (Ω; Rn). [ABSTRACT FROM AUTHOR]