Marcinkiewicz regularity for singular parabolic $p$-Laplace type equations with measure data

Abstract

We consider quasilinear parabolic equations with measurable coefficients when the right-hand side is a signed Radon measure with finite total mass, having $p$-Laplace type: $$u_t-\text{div}\mathbf{a}(Du,x,t)=\mu \text{in}\Omega \times (0,T)\subset {\mathbb{R}}^n\times \mathbb{R}.$$ In the singular range $\frac{2n}{n+1}<p\le 2-\frac{1}{n+1}$, we establish regularity estimates for the spatial gradient of solutions in the Marcinkiewicz spaces, under a suitable density condition of the right-hand side measure.

Major revision, 18 pages

References in corpus (4)

• Renormalized solutions of nonlinear parabolic equations with general measure data
• Diffuse measures and nonlinear parabolic equations
• Regularity estimates for singular parabolic measure data problems with sharp growth
• Gradient estimates for singular $p$-Laplace type equations with measure data