Convergence of nonlinear semigroups under nonpositive curvature
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Submission date: 01. Feb. 2013
published in: Transactions of the American Mathematical Society, 367 (2015) 6, p. 3929-3953
DOI number (of the published article): 10.1090/S0002-9947-2015-06087-5
MSC-Numbers: 46T20, 47H20, 58D07
Keywords and phrases: gradient flow, Mosco convergence, semigroup of nonexpansive maps
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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 convergence of the corresponding resolvents and 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., 2008].