Evolving inextensible and elastic curves with clamped ends under the second-order evolution equation in ℝ2

2018 ◽  
Vol 3 (1) ◽  
pp. 14-18 ◽  
Author(s):  
Chun-Chi Lin ◽  
Yang-Kai Lue
Keyword(s):  
Evolution Equation ◽  
Convergent Subsequence ◽  
Second Order ◽  
Plane Curves ◽  
Smooth Solutions ◽  
Fixed Position ◽  
Elastic Curves ◽  
Long Time Existence ◽  
Long Time ◽  
Time Existence

Abstract For any given C2-smooth initial open curves with fixed position and fixed tangent at the boundary points, we obtain the long-time existence of smooth solutions under the second-order evolution of plane curves. Moreover, the asymptotic limit of a convergent subsequence is an inextensible elastica.

scholarly journals A flow method for a generalization of $ L_{p} $ Christofell-Minkowski problem

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Boya Li ◽  
Hongjie Ju ◽  
Yannan Liu
Keyword(s):  
Initial Data ◽  
Curvature Flow ◽  
Minkowski Problem ◽  
Smooth Solutions ◽  
Flow Method ◽  
Long Time Existence ◽  
Long Time ◽  
Anisotropic Curvature ◽  
Time Existence

<p style='text-indent:20px;'>In this paper, a generalitzation of the <inline-formula><tex-math id="M2">\begin{document}$ L_{p} $\end{document}</tex-math></inline-formula>-Christoffel-Minkowski problem is studied. We consider an anisotropic curvature flow and derive the long-time existence of the flow. Then under some initial data, we obtain the existence of smooth solutions to this problem for <inline-formula><tex-math id="M3">\begin{document}$ c = 1 $\end{document}</tex-math></inline-formula>.</p>

scholarly journals Deforming a Convex Hypersurface by Anisotropic Curvature Flows

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
HongJie Ju ◽  
BoYa Li ◽  
YanNan Liu
Keyword(s):  
Principal Curvature ◽  
Support Function ◽  
Curvature Flow ◽  
Convex Hypersurface ◽  
Minkowski Problem ◽  
Smooth Solutions ◽  
Long Time Existence ◽  
Long Time ◽  
Anisotropic Curvature ◽  
Time Existence

AbstractIn this paper, we consider a fully nonlinear curvature flow of a convex hypersurface in the Euclidean 𝑛-space. This flow involves 𝑘-th elementary symmetric function for principal curvature radii and a function of support function. Under some appropriate assumptions, we prove the long-time existence and convergence of this flow. As an application, we give the existence of smooth solutions to the Orlicz–Christoffel–Minkowski problem.

scholarly journals Long-Time Existence for Semi-linear Beam Equations on Irrational Tori

2021 ◽  
Author(s):  
Joackim Bernier ◽  
Roberto Feola ◽  
Benoît Grébert ◽  
Felice Iandoli
Keyword(s):  
Long Time Existence ◽  
Beam Equations ◽  
Long Time ◽  
Time Existence

scholarly journals Long-time existence of the edge Yamabe flow

2019 ◽  
Vol 71 (2) ◽  
pp. 651-688 ◽  
Author(s):  
Eric BAHUAUD ◽  
Boris VERTMAN
Keyword(s):  
Yamabe Flow ◽  
Long Time Existence ◽  
Long Time ◽  
Time Existence

The elastic flow of curves on the sphere

2018 ◽  
Vol 3 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Anna Dall’Acqua ◽  
Tim Laux ◽  
Lin ◽  
Paola Pozzi ◽  
Adrian Spener
Keyword(s):  
Critical Points ◽  
Elastic Energy ◽  
Gradient Flow ◽  
Closed Curves ◽  
Long Time Existence ◽  
Long Time ◽  
Short Time ◽  
Time Existence

Abstract 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.

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