Output list
Journal article
Convergence of Sobolev gradient trajectories to elastica
Published 2025
Communications in analysis and geometry, 33, 5, 1199 - 1242
In this paper we study the H2(ds)-gradient flow for the modified elastic energy defined on closed immersed curves in. We prove the existence of a unique global-in-time solution to the flow and establish full convergence to elastica by way of a Łojasiewicz-Simon gradient inequality.
Journal article
A Sobolev gradient flow for the area-normalised Dirichlet energy of H1 maps
Published 2025
Advances in calculus of variations
In this article we study the H1-gradient flow for the energy E[X] given by the quotient of the Dirichlet energy and the signed enclosed area of an H1 map X : S -> R2. We prove that solutions with initially positive signed enclosed area exist eternally, and converge as t -> infinity to a (possibly multiply-covered) circle. In this way we recover an improved parametrised isoperimetric inequality for H1 maps.
Journal article
On the H1(dsγ)-Gradient Flow for the Length Functional
Published 2023
The Journal of geometric analysis, 33, 9, 297
In this article, we consider the length functional defined on the space of immersed planar curves...
Journal article
Representation formulae for higher order curvature flows
Published 2023
Journal of Differential Equations, 344, 1 - 43
In [25], Smoczyk showed that expansion of convex curves and hypersurfaces by the reciprocal of the harmonic mean curvature gives rise to a linear second order equation for the evolution of the support function, with corresponding representation formulae for solutions. In this article we consider L2(dθ)-gradient flows for a class of higher-order curvature functionals. These give rise to higher order linear parabolic equations for which we derive similar representation formulae for their solutions. Solutions exist for all time under natural conditions on the initial curve and converge exponentially fast in the smooth topology to multiply-covered circles. We consider both closed, embedded convex curves and closed, convex curves of higher rotation number. We give some corresponding remarks where relevant on open convex curves. We also consider corresponding ‘globally constrained’ flows which preserve the length or enclosed area of the evolving curve and a higher order approach to the Yau problem of evolving one convex planar curve to another. In an Appendix, we give some related second order results, including a version of the Yau problem for star-shaped curves.