Weak elastic energy of irregular curves

A weak notion of elastic energy for (not necessarily regular) rectifiable curves in any space dimension is proposed. Our $p$-energy is defined through a relaxation process, where a suitable $p$-rotation of inscribed polygonals is adopted. The discrete $p$-rotation we choose has a geometric flavor: a polygonal is viewed as an approximation to a smooth curve and hence its discrete curvature is spread out into a smooth density. For any exponent $p$ greater than one, the $p$-energy is finite if and only if the arc-length parameterization of the curve has a second order summability with the same growth exponent. In that case, moreover, the energy agrees with the natural extension of the integral of the $p$-th power of the scalar curvature. Finally, a comparison with other definitions of discrete curvatures is discussed.

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