Stability for parabolic quasi minimizers in metric measure spaces (Q1650248)

Summary: 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, \(p\geq2\), 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 [the authors et al., Potential Anal. 41, No. 3, 983--1004 (2014; Zbl 1327.35227)] to the setting of metric measure spaces.