Variational convergence over metric spaces

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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 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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Differential Geometry, Metric Geometry, 53C23; 49J45; 58E20

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http://salesiana.dossiersoluciones.com/handle/123456789/75252

Consulte el texto completo en el siguiente enlace:

https://arxiv.org/abs/math/0505430
 http://arxiv.org/abs/math/0505430

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arXiv