Stability for parabolic quasi minimizers in metric measure spaces
Yohei Fujishima, Jens Habermann · Rendiconti Lincei Matematica e Applicazioni · 2018
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.