Gradient estimates of parabolic Cauchy-Dirichlet problems on Morrey-Banach spaces (Q2091753)

The paper provides gradient estimates in Morrey type spaces for the solutions of the Cauchy-Dirichlet problem for divergence form linear parabolic equations with discontinuous coefficients. Precisely, weak solutions of the equation \[ u_t(x,t)-\sum_{\alpha,\beta=1}^n D_\alpha\big(a^{\alpha\beta}(x,t)D_\beta u(x,t)\big)=\sum_{\alpha=1}^n D_\alpha f^\alpha(x,t),\quad (x,t)\in\Omega_T \] are considered with zero data on the parabolic boundary of the cylinder \(\Omega_T=\Omega\times (0,T).\) Assuming that the boundary of \(\Omega\) is Reifenberg flat and the principal coefficients \(a^{\alpha\beta}\) are measurable with respect to one spatial variable and of small bounded mean oscillation in the remaining space and time variables, the author derives gradient estimates for the weak solutions in Lorentz-Morrey, Orlicz-Morrey and Morrey spaces with variable exponents.