We introduce for any exponent p>1 the p-curvature functional for rectifiable curves in the two-dimensional sphere. We prove that this functional is finite and agrees with the integral of the geodesic curvature raised to the power p on curves whose arc-length parameterization is in the Sobolev class W^{2,p}.
Weak elastic energy of rectifiable curves in the sphere / Mucci, Domenico; Saracco, Alberto; Sopio, Cristian. - In: PROCEEDINGS OF THE ROYAL SOCIETY OF LONDON. SERIES A. - ISSN 1364-5021. - 482:2336(2026), pp. 1-19. [10.1098/rspa.2025.0782]
Weak elastic energy of rectifiable curves in the sphere
Mucci, Domenico
;Saracco, Alberto;
2026
Abstract
We introduce for any exponent p>1 the p-curvature functional for rectifiable curves in the two-dimensional sphere. We prove that this functional is finite and agrees with the integral of the geodesic curvature raised to the power p on curves whose arc-length parameterization is in the Sobolev class W^{2,p}.
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