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

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 ut- Δ pu= μ with p∈ (1 , 2). The case when p∈(2-1n+1,2) 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∈(2nn+1,2-1n+1] if n≥ 2 and p∈(54,32] 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.

Duke Scholars

Author Hanye Zhu Mathematics