Global higher integrability of weak solutions of porous medium systems

Abstract

We establish higher integrability up to the boundary for the gradient of solutions to porous medium type systems, whose model case is given bywhere m > 1. More precisely, we prove that under suitable assumptions the spatial gradient D(|u| m−1 u) of any weak solution is integrable to a larger power than the natural power 2. Our analysis includes both the case of the lateral boundary and the initial boundary.

Global higher integrability for a doubly nonlinear parabolic system

“…To treat systems, and thus also signed solutions, another approach was used in [5]. Higher integrability for porous medium type systems up to the boundary is shown in [27]. For the singular case indicated by the condition m < 1, we refer to [7,15].…”

Stability for systems of porous medium type

“…To prove Theorem 2.7, one could repeat the same procedure as in the local setting, possibly using global higher integrability results in [15] to improve the convergence, at least in the degenerate case. However, we have chosen to use the local result.…”

A reverse Hölder inequality for the gradient of solutions to Trudinger’s equation

“…These results require a very careful analysis of the covering properties of the intrinsic cylinders relative to the solution. The ideas there, in part originating from [23], have later also been used to study systems of porous medium type [2] and global variants of the above mentioned problems [21].…”