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Convergence of nonlinear semigroups under nonpositive curvature


Author: Miroslav Bačák
Journal: Trans. Amer. Math. Soc. 367 (2015), 3929-3953
MSC (2010): Primary 46T20, 47H20, 58D07
DOI: https://doi.org/10.1090/S0002-9947-2015-06087-5
Published electronically: February 18, 2015
MathSciNet review: 3324915
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Abstract: The present paper is devoted to gradient flow semigroups of convex functionals on Hadamard spaces. We show that the Mosco convergence of a sequence of convex lsc functions implies the convergence of the corresponding resolvents and the convergence of the gradient flow semigroups. This extends the classical results of Attouch, Brezis and Pazy into spaces with no linear structure. The same method can be further used to show the convergence of semigroups on a sequence of spaces, which solves a problem of Kuwae and Shioya [Trans. Amer. Math. Soc. 360, no. 1, 2008].


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Additional Information

Miroslav Bačák
Affiliation: Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04 103 Leipzig, Germany
Address at time of publication: Telecom ParisTech, 37 rue Dareau, F-75014 Paris, France
Email: bacak@mis.mpg.de, bacak@telecom-paristech.fr

DOI: https://doi.org/10.1090/S0002-9947-2015-06087-5
Keywords: Convex function, gradient flow, {H}adamard space, Mosco convergence, resolvent, semigroup of nonexpansive maps, weak convergence
Received by editor(s): December 12, 2011
Received by editor(s) in revised form: December 14, 2012, January 10, 2013, and January 21, 2013
Published electronically: February 18, 2015
Additional Notes: The research leading to these results received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no. 267087
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.