Stability for parabolic quasi minimizers in metric measure spaces

Abstract

We are concerned with the stability property for parabolic quasi minimizers in metric measure spaces. More precisely we consider a doubling metric measure space $\mathcal{X}$ which supports a weak Poincaré inequality and a parabolic domain ${\Omega }_T=\Omega \times (0,T)$ on the product space $\mathcal{X}\times \mathbb{R}$, where $\Omega \subset \mathcal{X}$ is a domain whose boundary $\partial \Omega$ is regular in the sense that its complement satisfies a uniform capacity density condition. We then show that a parabolic $\mathcal{Q}$ quasi minimizer of the $p$ energy, $p\ge 2$, with fixed initial boundary data on the parabolic boundary of ${\Omega }_T$ is stable with respect to the variation of $\mathcal{Q}$ and $p$. The manuscript at hand is an extension of the result [7] to the setting of metric measure spaces.