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 polygons is adopted. The discrete p-rotation we choose has a geometric flavour: a polygon 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 1, the p-energy is finite if and only if the arc-length parametrization 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 pth power of the scalar curvature. Finally, a comparison with other definitions of discrete curvature is provided. This article is part of the theme issue 'Foundational issues, analysis and geometry in continuum mechanics'.
Weak elastic energy of irregular curves / Mucci, D.; Saracco, A.. - In: PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY OF LONDON SERIES A: MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES. - ISSN 1364-503X. - 381:2263(2023). [10.1098/rsta.2022.0370]
Weak elastic energy of irregular curves
Mucci D.
;Saracco A.
2023-01-01
Abstract
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 polygons is adopted. The discrete p-rotation we choose has a geometric flavour: a polygon 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 1, the p-energy is finite if and only if the arc-length parametrization 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 pth power of the scalar curvature. Finally, a comparison with other definitions of discrete curvature is provided. This article is part of the theme issue 'Foundational issues, analysis and geometry in continuum mechanics'.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.