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Gradient Flows in Abstract Metric Spaces: Evolution Variational Inequalities and Stability Muratori, Matteo


We study the main consequences of the existence of a Gradient Flow (GF for short), in the form of Evolution Variational Inequalities (EVI), in the very general framework of an abstract metric space. In particular, no volume measure is needed. The hypotheses on the functional associated with the GF are also very mild: we shall require at most completeness of the sublevels (no compactness assumption is made) and, for some convergence and stability results, approximate $\lambda$-convexity. The main results include: quantitative regularization properties of the flow (in terms e.g. of slope estimates and energy identities), discrete-approximation estimates of a minimizing-movement scheme and a stability theorem for the GF under suitable gamma-convergence-type hypotheses on a sequence of functionals approaching the limit functional. Existence of the GF itself is a quite delicate issue which requires some concavity-type assumptions on the metric, and will be addressed in a future project. This is a joint work with G. Savaré.

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