
Mathematics 2005
Variational convergence over metric spacesAbstract: We introduce a natural definition of $L^p$convergence of maps, $p \ge 1$, in the case where the domain is a convergent sequence of measured metric space with respect to the measured GromovHausdorff topology and the target is a GromovHausdorff convergent sequence. With the $L^p$convergence, we establish a theory of variational convergences. We prove that the Poincar\'e inequality with some additional condition implies the asymptotic compactness. The asymptotic compactness is equivalent to the GromovHausdorff compactness of the energysublevel sets. Supposing that the targets are $\CAT(0)$spaces, we study convergence of resolvents. As applications, we investigate the approximating energy functional over a measured metric space and convergence of energy functionals with a lower bound of Ricci curvature.
