We show well-posedness of the elastic flow of open curves with clamped boundary conditions. To show short time existence we prove that the linearised problem has the property of maximal L p -regularity and use the contraction principle to obtain the solution. Moreover, we show analyticity of the solution and its analytic dependency on the initial curve. With the developed methods we also prove long time existence of the flow if the initial curve is close to an elastica.1for some α > 0 and lim t→0 φ(t, ·) = φ 0 in C 2,α (I ).Furthermore the solution φ satisfies φ(t, ·)| ∂ I = ∂ x φ(t, ·)| ∂ I ≡ 0 for all t ∈ [0, T ] and depends analytically on the initial datum φ 0 .The proof of Theorem 1.2 will be given in Section 3. It consists of three steps: First we analyse the linearised equation and show that it has the property of maximal regularity, applying a general result from [7]. Afterwards, www.mn-journal.com
In this paper we consider the elastic energy for open curves in Euclidean space subject to clamped boundary conditions and obtain the Lojasiewicz-Simon gradient inequality for this energy functional. Thanks to this inequality we can prove that a (suitably reparametrized) solution to the associated L 2 -gradient flow converges for large time to an elastica, that is to a critical point of the functional.
We consider closed curves on the sphere moving by the L2-gradient flow of the elastic energy both with and without penalisation of the length and show short-time and long-time existence of the flow. Moreover, when the length is penalised, we prove sub-convergence to critical points.
bstract. We study Bakry-Émery curvature-dimension inequalities for non-local operators on the one-dimensional lattice and prove that operators with finite second moment have finite dimension. Moreover, we show that a class of operators related to the fractional Laplacian fails to have finite dimension and establish both positive and negative results for operators with sparsely supported kernels. Furthermore, a large class of operators is shown to have no positive curvature. The results correspond to CD inequalities on locally infinite graphs.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.