# Stability for parabolic quasi minimizers in metric measure spaces

### Yohei Fujishima

Shizuoka University, Hamamatsu, Japan### Jens Habermann

Universität Erlangen-Nürnberg, Germany

## 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 $X$ which supports a weak Poincaré inequality and a parabolic domain $Ω_{T}=Ω×(0,T)$ on the product space $X×R$, where $Ω⊂X$ is a domain whose boundary $∂Ω$ is regular in the sense that its complement satisfies a uniform capacity density condition. We then show that a parabolic $Q$ quasi minimizer of the $p$ energy, $p≥2$, with fixed initial boundary data on the parabolic boundary of $Ω_{T}$ is stable with respect to the variation of $Q$ and $p$. The manuscript at hand is an extension of the result [7] to the setting of metric measure spaces.

## Cite this article

Yohei Fujishima, Jens Habermann, Stability for parabolic quasi minimizers in metric measure spaces. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 29 (2018), no. 2, pp. 343–376

DOI 10.4171/RLM/810