A regularized gradient flow for the <i>p</i> -elastic energy
Simon Blatt, Christopher Hopper, Nicole Vorderobermeier
Abstract
Abstract We prove long-time existence for the negative <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:math> {L}^{2} -gradient flow of the p -elastic energy, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:math> p\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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:math> {L}^{2} -norm of the normal gradient of the curvature <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>κ</m:mi> </m:math> \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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>W</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> </m:math> {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.