Regularity estimates for singular parabolic measure data problems with sharp growth

We prove global gradient estimates for parabolic $p$-Laplace type equations with measure data, whose model is $u_t - \textrm{div} \left(|Du|^{p-2} Du\right) = μ\quad \textrm{in} \ Ω\times (0,T) \subset \mathbb{R}^n \times \mathbb{R},$ where $μ$ is a signed Radon measure with finite total mass. We consider the singular case $\frac{2n}{n+1} <p \le 2-\frac{1}{n+1}$ and give possibly minimal conditions on the nonlinearity and the boundary of $Ω$, which guarantee the regularity results for such measure data problems.

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