A regularized gradient flow for the p-elastic energy

Abstract

We prove long-time existence for the negative L2{L}^{2}-gradient flow of the p-elastic energy, p≥2p\ge 2, with an additive positive multiple of the length of the curve. To achieve this result, we regularize the energy by cutting off the degeneracy at points with vanishing curvature and add a small multiple of a higher order energy, namely, the square of the L2{L}^{2}-norm of the normal gradient of the curvature κ\kappa . Long-time existence is proved for the gradient flow of these new energies together with the smooth subconvergence of the evolution equation’s solutions to critical points of the regularized energy in W2,p{W}^{2,p}. We then show that the solutions to the regularized evolution equations converge to a weak solution of the negative gradient flow of the p-elastic energies. These latter weak solutions also subconverge to critical points of the p-elastic energy.

Keywords

• geometric evolution equation
• degenerate evolution equation
• fourth order
• p-elastic curves
• 53c44
• 53a04