Calculus of Variations and Geometric Measure Theory
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N. Gigli - C. Rigoni

Partial derivatives in the nonsmooth setting

created by rigoni on 07 Dec 2020

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Submitted Paper

Inserted: 7 dec 2020
Last Updated: 7 dec 2020

Year: 2020

Abstract:

We study partial derivatives on the product of two metric measure structures, in particular in connection with calculus via modules as proposed by the first named author. Our main results are: i) The extension to this non-smooth framework of Schwarz’s theorem about symmetry of mixed second derivatives ii) A quite complete set of results relating the property f ∈ W2,2(X × Y) on one side with that of f(·,y) ∈ W2,2(X) and f(x,·) ∈ W2,2(Y) for a.e. y,x respectively on the other. Here X, Y are RCD spaces so that second order Sobolev spaces are well defined. These results are in turn based upon the study of Sobolev regularity, and of the under- lying notion of differential, for a map with values in a Hilbert module.


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