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Transactions of the American Mathematical Society

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Variational convergence over metric spaces


Authors: Kazuhiro Kuwae and Takashi Shioya
Journal: Trans. Amer. Math. Soc. 360 (2008), 35-75
MSC (2000): Primary 53C23; Secondary 49J45, 58E20
DOI: https://doi.org/10.1090/S0002-9947-07-04167-0
Published electronically: August 22, 2007
MathSciNet review: 2341993
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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 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 $ \operatorname{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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Additional Information

Kazuhiro Kuwae
Affiliation: Department of Mathematics, Faculty of Education, Kumamoto University, Kuma- moto, 860-8555, Japan
Email: kuwae@gpo.kumamoto-u.ac.jp

Takashi Shioya
Affiliation: Mathematical Institute, Tohoku University, Sendai 980-8578, Japan
Email: shioya@math.tohoku.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-07-04167-0
Keywords: Measured metric space, $L^p$-mapping space, Gromov-Hausdorff convergence, Mosco convergence, $\Gamma$-convergence, asymptotic compactness, the Poincar\'e inequality, $\operatorname{CAT}(0)$-space, harmonic map, resolvent, spectrum
Received by editor(s): July 7, 2005
Published electronically: August 22, 2007
Additional Notes: The first author was partially supported by a Grant-in-Aid for Scientific Research No. 16540201 from the Ministry of Education, Science, Sports and Culture, Japan
The second author was partially supported by a Grant-in-Aid for Scientific Research No. 14540056 from the Ministry of Education, Science, Sports and Culture, Japan
Article copyright: © Copyright 2007 American Mathematical Society

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