Global Marcinkiewicz estimates for nonlinear parabolic equations with nonsmooth coefficients

Abstract

Consider the parabolic equation with measure data \begin{equation*} \left\{ \begin{aligned} &u_t-{\rm div} \mathbf{a}(D u,x,t)=μ&\text{in}& \quad Ω_T, &u=0 \quad &\text{on}& \quad \partial_pΩ_T, \end{aligned}\right. \end{equation*} where $\Omega$ is a bounded domain in ${\mathbb{R}}^n$, ${\Omega }_T=\Omega \times (0,T)$, ${\partial }_p{\Omega }_T=(\partial \Omega \times (0,T))\cup (\Omega \times \{0\})$, and $\mu$ is a signed Borel measure with finite total mass. Assume that the nonlinearity $\mathbf{a}$ satisfies a small BMO-seminorm condition, and $\Omega$ is a Reifenberg flat domain. This paper proves a global Marcinkiewicz estimate for the SOLA (Solution Obtained as Limits of Approximation) to the parabolic equation.

23 pages, to appear in Ann. Sc. Norm. Super. Pisa. arXiv admin note: text overlap with arXiv:1702.06202