A Willmore-Helfrich $L^2$-flow of curves with natural boundary conditions

Abstract

We consider regular open curves in R n with fixed boundary points and moving according to the L 2 -gradient flow for a generalisation of the Helfrich functional. Natural boundary conditions are imposed along the evolution. More precisely, at the boundary the curvature vector is equal to the normal projection of a fixed given vector. A long-time existence result together with subconvergence to critical points is proven.

Short time existence for the elastic flow of clamped curves

“…The elastic energy is the one-dimensional analogue of the Willmore energy and has been studied extensively, see [14] and the references therein. Long time existence of the elastic flow for λ > 0 is due to [8] for closed curves, for open curves with clamped boundary conditions due to [14], for curvature-dependent boundary conditions due to [5]. All works mentioned only sketch the necessary arguments for the short time existence.…”

Evolution of open elastic curves in ℝn subject to fixed length and natural boundary conditions

“…The following results are adaptations to the present setting and notation of those used in [2] for closed curves and in [3] and [1] for open ones. …”

“…Since for any time ∈ (0, ) we have that From here the proof proceeds as in [2, Theorem 3.3]: however, for the ease of exposition we will sometimes refer the reader to results in [1], which are adaptation to the case of open curves of arguments given in [2]. We sketch here the main ideas.…”

“…The latter can be called a real stroke of luck. Indeed, if one considers different types of boundary conditions, new methods of proof must be found in order to be able to deal with the boundary terms: example in this direction are the works [3] and [1]. For this reason the extension of the results presented here to the case of clamped boundary conditions will be treated by the authors in a subsequent work.…”

“…, ⃗ , the product ⃗ 1 * ⋅ ⋅ ⋅ * ⃗ defines for even a function given by ⟨ ⃗ 1 , ⃗ 2 ⟩ ⋅ ⋅ ⋅ ⟨ ⃗ −1 , ⃗ ⟩, while for odd it defines a normal vector field ⟨ ⃗ 1 , ⃗ 2 ⟩ ⋅ ⋅ ⋅ ⟨ ⃗ −2 , ⃗ −1 ⟩ ⃗ . Following the notation adopted in [1], for ⃗ a normal vector field, , ( ⃗ ) denotes any linear combination of terms of type…”