Abstract. The study of the Sobolev space over weighted Euclidean spaces
(i.e. equipped with an arbitrary Radon measure) has been initiated in
the late nineties, motivated by various applications in the field of the
Calculus of Variations. Two important approaches were introduced by
Bouchitté-Buttazzo-Seppecher and Zhikov, both relying upon a notion of
'Sobolev tangent bundle'. In this talk, we will prove the equivalence
of these two theories, by employing a third notion of Sobolev space
due to Ambrosio-Gigli-Savaré, which comes from the more general setting
of analysis on metric measure spaces. Moreover, we will investigate
the relation between the above-mentioned 'Sobolev tangent bundle' and
the 'Lipschitz tangent bundle' introduced by Alberti-Marchese. Finally,
we will provide necessary and sufficient conditions for the Sobolev/Lipschitz
tangent bundles to have full rank.
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