Evolution equations with lack of convexity

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      This is the first of a series of papers devoted to a thorough analysis of the class of gradient flows in metric spaces that can be characterized by Evolution Variational Inequalities (EVI, in short). Under the initial impulse of De Giorgi, Degiovanni, Marino, Tosques [37,38,61], the abstract theory has been extended towards two main directions: a relaxation of the convexity assumptions on ϕ (see e.g. [67,80]) and a broadening of the structure of the ambient space, from Hilbert to Banach spaces (for the theory of doubly nonlinear evolution equations, see e.g. [15,28]) or to more general metric and topological spaces [36]. It is remarkable that the original approach by De Giorgi and his collaborators encompasses both these directions.

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