Weighted Orlicz estimates for general nonlinear parabolic equations over nonsmooth domains

Abstract

We introduce a parabolic analogue of Muckenhoupt weights to study optimal weighted regularity in Orlicz spaces for a general nonlinear parabolic problem of p-Laplacian-type in divergence form over a nonsmooth domain. Assuming that the nonlinearity is measurable with respect to the time variable and has a small bounded mean oscillation (BMO) with respect to the spatial variables, that the lateral boundary of the parabolic cylinder is δ-Reifenberg flat and that the associated weight belongs to a suitable parabolic Muckenhoupt class, we obtain a global gradient estimate for such a nonlinear parabolic problem by essentially proving that the gradient of the weak solution is as globally integrable as the nonhomogeneous term in the weighted Orlicz space. Our results extend the existing regularity estimates in Lebesgue spaces to weighted Orlicz spaces.