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We develop a general mathematical framework for variational problems where the unknown function assumes values in the space of probability measures on some metric space. We study suitable weak and strong topologies and define a total variation seminorm for functions taking values in a Banach space. For a class of variational problems based on this formulation, we prove existence and point out connections to the Kantorovich-Rubinstein norm and optimal transport.
This presentation is part of Minisymposium “MS31 - Variational Approaches for Regularizing Nonlinear Geometric Data (3 parts)”
organized by: Martin Storath (Universität Heidelberg) , Martin Holler (École Polytechnique, Université Paris Saclay) , Andreas Weinmann (Hochschule Darmstadt) .