## $$p$$-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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