Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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D. Di Donato

Intrinsic Cheeger energy for the intrinsically Lipschitz constants

created by didonato on 01 Jun 2022



Inserted: 1 jun 2022
Last Updated: 1 jun 2022

Year: 2022

ArXiv: 2205.15851 PDF


Recently, in the metric spaces, Le Donne and the author introduced the so-called intrinsically Lipschitz sections. The main aim of this note is to adapt Cheeger theory for the classical Lipschitz constants in our new context. More precisely, we define the intrinsic Cheeger energy from $L^2(Y)$ to $[1,+\infty],$ where $(Y,d_Y,\mm)$ is a metric measure space with $Y\subset \R$ and we characterize it in terms of a suitable notion of relaxed slope. In order to get this result, in more general context, we establish some properties of the intrinsically Lipschitz constants like the Leibniz formula, the product formula and the upper semicontinuity of the asymptotic in\-trin\-si\-cally Lipschitz constant. Finally, adapting the concept of relaxed slope in our intrinsic context, we define the so-called intrinsic relaxed slope in strong sense and we show that this new class is a vector space over $\R.$

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