Gradient regularity results in mixed local and nonlocal problems

The talk is devoted to the problem studied in cooperation with Professor Atsushi Tachikawa. We treat the regularity problem for minimizers u(x) of quadratic and nonquadratic growth functional having integrand A(x, u, Du). We point out that it is assumed only that A(x,u, p) is in the Vanishing Mean Oscillation class (as a function of x). Namely, it is not assumed the continuity of A(x, u, p) with respect to x. There are treated partial regularity and global regularity of the minimizer u.