Local higher integrability for parabolic quasiminimizers in metric spaces

Abstract

Using purely variational methods, we prove in metric measure spaces local higher integrability for minimal p-weak upper gradients of parabolic quasiminimizers related to the heat equation. We assume the measure to be doubling and the underlying space to be such that a weak Poincaré inequality is supported. We define parabolic quasiminimizers in the general metric measure space context, and prove an energy type estimate. Using the energy estimate and properties of the underlying metric measure space, we prove a reverse Hölder inequality type estimate for minimal \(p\)-weak upper gradients of parabolic quasiminimizers. Local higher integrability is then established based on the reverse Hölder inequality, by using a modification of Gehring’s lemma.