Self-improving property of degenerate parabolic equations of porous medium-type

Abstract

We show that the gradient of solutions to degenerate parabolic equations of porous medium-type satisfies a reverse Hölder inequality in suitable intrinsic cylinders. We modify the by-now classical Gehring lemma by introducing an intrinsic Calderón-Zygmund covering argument, and we are able to prove local higher integrability of the gradient of a proper power of the solution u.

Self-improving property of the fast diffusion equation

“…In this paper we are interested in the order of integrability of |Du m |; we will deal only with the range m ∈ (n − 2) + n + 2 , 1 , (1.4) (the case m > 1 has already been dealt with in [GS16]), and we will study a general class of equations which have the same structure as (PME). Notice that we are not considering the full interval m ∈ (0, 1): the lower bound on m is quite typical, when dealing with regularity issues for the singular porous medium equation (see, for example, the discussion in [DGV, Chapter 6, Paragraph 21]).…”

“…A second novelty of [GS16] is in Calderón-Zygmund covering, where we used cylinders which are intrinsically scaled with respect to what seems the natural quantity here, namely u m−1 , where u is the solution. In other terms, we considered cylinders of the type Q θρ 2 ,ρ whose space-time scaling is adapted to the solution u via the coupling…”

“…Our estimates are satisfied for a general class of growth assumptions on the non linearity. In this way, we extend the theory for m ≥ 1 (see [GS16] in the list of references) to the singular case. In particular, an intrinsic metric that depends on the solution itself is introduced for the singular regime.…”

Higher integrability for the singular porous medium system

“…Construction of a non-uniform system of cylinders. The following construction of a non-uniform system of cylinders is similar to the one in [11,19]. Let z o ∈ Q 2R .…”

Global higher integrability of weak solutions of porous medium systems

“…Both of these coupling conditions have to be taken into account for the construction of a system of sub-intrinsic cylinders as in [15]. In fact, when considering a point z o close to the lateral boundary, it is not clear a priori if the mentioned construction yields a cylinder for which the doubled cylinder Q (θ) 2 (z o ) touches the boundary or not.…”