Global Lorentz estimates for nonlinear parabolic equations on nonsmooth domains

Abstract

Consider the nonlinear parabolic equation in the form $$u_t-\mathrm{div}\mathbf{a}(Du,x,t)=\mathrm{div}(|F{|}^{p-2}F)\text{in}\Omega \times (0,T),$$ where $T>0$ and $\Omega$ is a Reifenberg domain. We suppose that the nonlinearity $\mathbf{a}(\xi ,x,t)$ has a small BMO norm with respect to $x$ and is merely measurable and bounded with respect to the time variable $t$. In this paper, we prove the global Calderón-Zygmund estimates for the weak solution to this parabolic problem in the setting of Lorentz spaces which includes the estimates in Lebesgue spaces. Our global Calderón-Zygmund estimates extend certain previous results to equations with less regularity assumptions on the nonlinearity $\mathbf{a}(\xi ,x,t)$ and to more general setting of Lorentz spaces.

21 pages, to appear in CV&PDEs. arXiv admin note: text overlap with arXiv:1702.06200