Weak elastic energy of rectifiable curves in the sphere

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 $W^{2,p}$.

21 pages, 2 figures