Gradient estimates for mixed local and nonlocal parabolic problems with measure data

Abstract

We obtain a nonlinear Calderón-Zygmund type estimate for the spatial gradient of SOLA (solutions obtained as limits of approximations) to a mixed local and nonlocal parabolic problem involving measure data. Specifically, we consider a non-homogeneous problem with a source term as a Radon measure, where the leading nonlinear local operator has linear growth and the nonlinearity is allowed to be merely measurable. Moreover, the nonlocal operator is of the type fractional Laplacian with only measurable kernel coefficients. Under some smallness assumption on the local operator, we prove that the gradient of SOLA has the same integrability as that of a fractional maximal function of order 1 of the Radon measure. The proof relies on the use of a new type of maximal functions and novel parabolic tail estimates, which to the best of our knowledge, have not been observed in the previous works. Furthermore, to complement the aforementioned gradient estimates, we prove the existence of SOLA to an initial value problem and derive some gradient estimates with respect to the L1-norm.