\(p\)-energy of a curve on LIP-manifolds and on general metric spaces. (English) Zbl 0956.53051

The concept of \(p\)-energy (\(p\geq 1\)) of a curve can be generalized in a general metric space \((S,\sigma)\) to three types of functionals \(E_h(\sigma,p)\) (\(h=1,2,3\)). The authors determine conditions which provide the coincidence of these functionals. They introduce a suitable notion of asymptotically equal generalized distances, and prove that for two such distances \(\sigma\) and \(\rho\), it follows that \(E_h(\sigma,p)(\gamma)=E_h(\rho,p)(\gamma)\) for any \(p\geq 1\), when the curve \(\gamma\) of \(S\) has finite energy for some \(p_0>1\). At the end, a meaningful example is given, considering a particular generalized distance \(\sigma=\sigma_r\) (\(r\geq 1\)), defined on the set of the Lebesgue measurable parts in \({\mathbb R}^n\), which is used in the study of minimizing motions.


53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
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