Variational convergence over metric spaces

2005/05/20 by Kazuhiro Kuwae, Kuwae, Kazuhiro, Takashi Shioya +1 · 1 citation
Mathematics · #49J45 #53C23 #58E20 #Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG) #math.DG #math.MG #msc:49J45 #msc:53C23 #msc:58E20

paper · pdf · doi:10.48550/arxiv.math/0505430

arxiv created 2005/05/20 · arxiv updated 2009/12/01

Abstract

We introduce a natural definition of L^p-convergence of maps, p ≥ 1, in the case where the domain is a convergent sequence of measured metric space with respect to the measured Gromov-Hausdorff topology and the target is a Gromov-Hausdorff convergent sequence. With the L^p-convergence, we establish a theory of variational convergences. We prove that the Poincaré inequality with some additional condition implies the asymptotic compactness. The asymptotic compactness is equivalent to the Gromov-Hausdorff compactness of the energy-sublevel 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.

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