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 Relative Articles Variational convergence of gradient flows and rate-independent evolutions in metric spaces On the relaxation of variational integrals in metric Sobolev spaces Probabilistic convergence spaces and generalized metric spaces A notion of weak convergence in metric spaces The variational capacity with respect to nonopen sets in metric spaces Entropy and its variational principle for noncompact metric spaces Convergence of Manifolds and Metric Spaces with Boundary Flat convergence for integral currents in metric spaces Convergence to Compact Sets of Inexact Orbits of Nonexpansive Mappings in Banach and Metric Spaces Convergence to Compact Sets of Inexact Orbits of Nonexpansive Mappings in Banach and Metric Spaces More...
Mathematics  2005

# Variational convergence over metric spaces

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Abstract:

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 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 Poincar\'e 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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