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\(p\)-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.

MSC:

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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