Gradient Estimates of ω-Minimizers to Double Phase Variational Problems with Variable Exponents

We are concerned with an optimal regularity for ω-minimizers to double phase variational problems with variable exponents where the associated energy density is allowed to be discontinuous. We identify basic structure assumptions on the density for the absence of Lavrentiev phenomenon and higher integrability. Moreover, we establish a local Calderón–Zygmund theory for such generalized minimizers under minimal regularity requirements regarding such double phase functionals to the frame of Lebesgue spaces with variable exponents.