Gradient estimates for singular parabolic $p$-Laplace type equations with measure data

Abstract

We are concerned with gradient estimates for solutions to a class of singular quasilinear parabolic equations with measure data, whose prototype is given by the parabolic $p$-Laplace equation $u_t-{\Delta }_{p}u=\mu$ with $p\in (1,2)$. The case when $p\in (2-\frac{1}{n+1},2)$ were studied in [15]. In this paper, we extend the results in [15] to the open case when $p\in (\frac{2n}{n+1},2-\frac{1}{n+1}]$ if $n\ge 2$ and $p\in (\frac{5}{4},\frac{3}{2}]$ if $n=1$. More specifically, in a more singular range of $p$ as above, we establish pointwise gradient estimates via linear parabolic Riesz potential and gradient continuity results via certain assumptions on parabolic Riesz potential.

38 pages, comments are welcome