scholarly journals Anisotropic Curvature Flow of Immersed Networks

2021 ◽  
Author(s):  
Heiko Kröner ◽  
Matteo Novaga ◽  
Paola Pozzi
Keyword(s):  
Triple Junction ◽  
Curvature Flow ◽  
The Other ◽  
Anisotropic Curvature ◽  
Maximal Time

AbstractWe consider motion by anisotropic curvature of a network of three curves immersed in the plane meeting at a triple junction and with the other ends fixed. We show existence, uniqueness and regularity of a maximal geometric solution and we prove that, if the maximal time is finite, then either the length of one of the curves goes to zero or the $$L^2$$ L 2 -norm of the anisotropic curvature blows up.

scholarly journals Nonlinear stability of stationary solutions for curvature flow with triple junction

2009 ◽  
Vol 38 (4) ◽  
pp. 721-769 ◽  
Author(s):  
Harald GARCKE ◽  
Yoshihito KOHSAKA ◽  
Daniel ŠEVČOVIČ
Keyword(s):  
Nonlinear Stability ◽  
Triple Junction ◽  
Curvature Flow ◽  
Stationary Solutions

scholarly journals Remarks on the self-shrinking Clifford torus

2020 ◽  
Vol 2020 (765) ◽  
pp. 139-170
Author(s):  
Christopher G. Evans ◽  
Jason D. Lotay ◽  
Felix Schulze
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
The Self ◽  
The Other ◽  
Local Entropy ◽  
Clifford Torus ◽  
Infinitesimal Deformations ◽  
Rigid Motions ◽  
The One

AbstractOn the one hand, we prove that the Clifford torus in {\mathbb{C}^{2}} is unstable for Lagrangian mean curvature flow under arbitrarily small Hamiltonian perturbations, even though it is Hamiltonian F-stable and locally area minimising under Hamiltonian variations. On the other hand, we show that the Clifford torus is rigid: it is locally unique as a self-shrinker for mean curvature flow, despite having infinitesimal deformations which do not arise from rigid motions. The proofs rely on analysing higher order phenomena: specifically, showing that the Clifford torus is not a local entropy minimiser even under Hamiltonian variations, and demonstrating that infinitesimal deformations which do not generate rigid motions are genuinely obstructed.

Reactive wetting of polycrystalline TiC by molten Zr55Cu30Al10Ni5 metallic glass alloy

2009 ◽  
Vol 24 (7) ◽  
pp. 2420-2427 ◽  
Author(s):  
Ping Shen ◽  
Qiaoli Lin ◽  
Qichuan Jiang ◽  
Hidetoshi Fujii ◽  
Kiyoshi Nogi
Keyword(s):  
Metallic Glass ◽  
Reaction Layer ◽  
Triple Junction ◽  
Sessile Drop ◽  
Contact Angles ◽  
Sessile Drop Method ◽  
The Other ◽  
Reactive Wetting ◽  
Exponential Functions ◽  
Glass Alloy

The isotherm wetting and spreading behaviors of polycrystalline TiC by molten Zr55Cu30Al10Ni5 alloy were investigated at 1133 to 1253 K in a vacuum by using a modified sessile drop method. The system displays good wettability with the initial and final stable contact angles of 44 to 50° and 10 to 14°, respectively, mildly depending on the temperature. The spreading kinetics follows well defined exponential functions and could be described by a molecular dynamic model. On the other hand, a ZrC reaction layer was formed at the interface. The wetting was primarily promoted by the Zr adsorption at the triple junction and then by the formation of the reaction layer.

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 Ray Tracing Comparison between Triple-Junction and Four-Junction Solar Cells in PMMA Fresnel-Based High-CPV Units

Energies ◽  
2018 ◽  
Vol 11 (9) ◽  
pp. 2455 ◽  
Author(s):  
Juan Ferrer-Rodríguez ◽  
Alvaro Valera ◽  
Eduardo Fernández ◽  
Florencia Almonacid ◽  
Pedro Pérez-Higueras
Keyword(s):  
Solar Cells ◽  
Ray Tracing ◽  
Triple Junction ◽  
Optical Element ◽  
The Other ◽  
Optical Systems ◽  
Single Lens ◽  
The One ◽  
Optical Concentrator ◽  
Spectral Responses

The recent development of wafer bonded four-junction concentrator solar cells (FJSCs) with record efficiency among all the existent photovoltaic (PV) cells offers new possibilities for improving the High Concentrator PV (HCPV) technology. However, the concentrator optical systems utilized in HCPV modules may have to be adapted to the new requirements of FJSC in order to properly take advantage of the increased number of p-n junctions. This research theoretically compares two identical optical concentrator systems, a Frensel lens plus a kind of refractive SILO (SIngle-Lens-Optical element) secondary (both made of PMMA, poly(methyl methacrylate)), which are equipped with a typical triple-junction concentrator solar cell (TJSC) in the one case, and with an FJSC in the other case. Both HCPV units are analyzed through ray tracing optical simulations applying an exhaustive optical modelling that takes into account the spectral responses of the different subcells within the multi-junction cells. The HCPV unit with the FJSC and PMMA SOE (secondary optical element) shows much less efficiency than that with the TJSC due to the light absorption through the PMMA SOE in the wavelength range of the bottom subcell. Therefore, PMMA SOEs may be not appropriate for FJSC in general.

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 On a curvature flow in a band domain with unbounded boundary slopes

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Lixia Yuan ◽  
Wei Zhao
Keyword(s):  
Curvature Flow ◽  
Gradient Estimates ◽  
Normal Vector ◽  
Normal Velocity ◽  
Unit Normal Vector ◽  
Contact Points ◽  
Anisotropic Curvature ◽  
Robin Boundary ◽  
Zero Number ◽  
Boundary Slopes

<p style='text-indent:20px;'>This paper is devoted to an anisotropic curvature flow of the form <inline-formula><tex-math id="M1">\begin{document}$ V = A(\mathbf{n})H + B(\mathbf{n}) $\end{document}</tex-math></inline-formula> in a band domain <inline-formula><tex-math id="M2">\begin{document}$ \Omega : = [-1,1]\times {\mathbb{R}} $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ \mathbf{n} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ V $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ H $\end{document}</tex-math></inline-formula> denote respectively the unit normal vector, normal velocity and curvature of a graphic curve <inline-formula><tex-math id="M6">\begin{document}$ \Gamma_t $\end{document}</tex-math></inline-formula>. We require that the curve <inline-formula><tex-math id="M7">\begin{document}$ \Gamma_t $\end{document}</tex-math></inline-formula> contacts <inline-formula><tex-math id="M8">\begin{document}$ \partial \Omega $\end{document}</tex-math></inline-formula> with slopes equaling to the heights of the contact points (which corresponds to a kind of Robin boundary conditions). In spite of the unboundedness of the boundary slopes, we are able to obtain the <i>uniform interior gradient estimates</i> for the solutions by using the zero number argument. Furthermore, when <inline-formula><tex-math id="M9">\begin{document}$ t\to \infty $\end{document}</tex-math></inline-formula>, we show that <inline-formula><tex-math id="M10">\begin{document}$ \Gamma_t $\end{document}</tex-math></inline-formula> converges to a traveling wave with cup-shaped profile and <i>infinite</i> boundary slopes in the <inline-formula><tex-math id="M11">\begin{document}$ C^{2,1}_{\rm{loc}} ((-1,1)\times {\mathbb{R}}) $\end{document}</tex-math></inline-formula>-topology.</p>

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