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)=\mu&\text{in}& \quad \Omega_T, &u=0 \quad &\text{on}& \quad \partial_p\Omega_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 ${\bf a}$ 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.