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.
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The authors would like to thank the referee for his careful reading and very helpful comments.
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Communicated by N. S. Trudinger.
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H. Dong was partially supported by a Simons fellowship grant no. 007638, the NSF under agreement DMS-2055244, and the Charles Simonyi Endowment at the Institute of Advanced Study. H. Zhu was partially supported by the NSF under agreement DMS-2055244.
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Dong, H., Zhu, H. Gradient estimates for singular parabolic p-Laplace type equations with measure data. Calc. Var. 61, 86 (2022). https://doi.org/10.1007/s00526-022-02189-5
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DOI: https://doi.org/10.1007/s00526-022-02189-5