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.

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 Orlicz–Minkowski flows

2021 ◽  
Vol 60 (1) ◽  
Author(s):  
Paul Bryan ◽  
Mohammad N. Ivaki ◽  
Julian Scheuer
Keyword(s):  
Final Section ◽  
Gauss Curvature ◽  
Stationary Solutions ◽  
Existence Results ◽  
Minkowski Problem ◽  
Parabolic Approximation ◽  
Long Time Existence ◽  
Long Time ◽  
And Behavior ◽  
Time Existence

AbstractWe study the long-time existence and behavior for a class of anisotropic non-homogeneous Gauss curvature flows whose stationary solutions, if they exist, solve the regular Orlicz–Minkowski problems. As an application, we obtain old and new existence results for the regular even Orlicz–Minkowski problems; the corresponding $$L_p$$ L p version is the even $$L_p$$ L p -Minkowski problem for $$p>-n-1$$ p > - n - 1 . Moreover, employing a parabolic approximation method, we give new proofs of some of the existence results for the general Orlicz–Minkowski problems; the $$L_p$$ L p versions are the even $$L_p$$ L p -Minkowski problem for $$p>0$$ p > 0 and the $$L_p$$ L p -Minkowski problem for $$p>1$$ p > 1 . In the final section, we use a curvature flow with no global term to solve a class of $$L_p$$ L p -Christoffel–Minkowski type problems.

scholarly journals Minkowski inequalities and constrained inverse curvature flows in warped spaces

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Julian Scheuer
Keyword(s):  
Mean Curvature Flow ◽  
Broad Class ◽  
Higher Dimensions ◽  
Curvature Flow ◽  
Radial Coordinate ◽  
Inverse Mean Curvature Flow ◽  
Long Time Existence ◽  
Long Time ◽  
Smooth Convergence ◽  
Time Existence

AbstractThis paper deals with locally constrained inverse curvature flows in a broad class of Riemannian warped spaces. For a certain class of such flows, we prove long-time existence and smooth convergence to a radial coordinate slice. In the case of two-dimensional surfaces and a suitable speed, these flows enjoy two monotone quantities. In such cases, new Minkowski type inequalities are the consequence. In higher dimensions, we use the inverse mean curvature flow to obtain new Minkowski inequalities when the ambient radial Ricci curvature is constantly negative.

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