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 $$p\in \big (2-\frac{1}{n+1},2\big )$$ were studied in Kuusi and Mingione (Ann Sc Norm Super Pisa Cl Sci 5 12(4):755–822, 2013). In this paper, we extend the results in Kuusi and Mingione (2013) to the open case when $$p\in \big (\frac{2n}{n+1},2-\frac{1}{n+1}\big ]$$ 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.