Remarks on the generalized Cauchy-Dirichlet problem for graph mean curvature flow with driving force

Abstract A general purely crystalline mean curvature flow equation with a nonuniform driving force term is considered. The unique existence of a level set flow is established when the driving force term is continuous and spatially Lipschitz uniformly in time. By introducing a suitable notion of a solution a comparison principle of continuous solutions is established for equations including the level set equations. An existence of a solution is obtained by stability and approximation by smoother problems. A necessary equi-continuity of approximate solutions is established. It should be noted that the value of crystalline curvature may depend not only on the geometry of evolving surfaces but also on the driving force if it is spatially inhomogeneous.

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On obstacle problem for mean curvature flow with driving force

Geometric Flows ◽

10.1515/geofl-2019-0002 ◽

2019 ◽

Vol 4 (1) ◽

pp. 9-29

Author(s):

Yoshikazu Giga ◽

Hung V. Tran ◽

Longjie Zhang

Keyword(s):

Mean Curvature ◽

Driving Force ◽

Obstacle Problem ◽

Mean Curvature Flow ◽

Boundary Regularity ◽

Curvature Flow ◽

Convergence Result ◽

Large Time Behavior ◽

Time Behavior ◽

Full Characterization

Abstract In this paper, we study an obstacle problem associated with the mean curvature flow with constant driving force. Our first main result concerns interior and boundary regularity of the solution. We then study in details the large time behavior of the solution and obtain the convergence result. In particular, we give full characterization of the limiting profiles in the radially symmetric setting.

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Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow

Open Mathematics ◽

10.1515/math-2020-0090 ◽

2020 ◽

Vol 18 (1) ◽

pp. 1518-1530

Author(s):

Xuesen Qi ◽

Ximin Liu

Keyword(s):

Laplace Operator ◽

Mean Curvature ◽

Mean Curvature Flow ◽

Curvature Flow ◽

Second Fundamental Form ◽

Coefficient Function ◽

First Eigenvalue ◽

Initial Hypersurface ◽

The Mean ◽

The Laplace Operator

Abstract In this paper, we discuss the monotonicity of the first nonzero eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow (MCF). By imposing conditions associated with the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced MCF, and some special pinching conditions on the second fundamental form of the initial hypersurface, we prove that the first nonzero closed eigenvalues of the Laplace operator and the p-Laplace operator are monotonic under the forced MCF, respectively, which partially generalize Mao and Zhao’s work. Moreover, we give an example to specify applications of conclusions obtained above.

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Mean curvature flow with free boundary outside a hypersphere

Transactions of the American Mathematical Society ◽

10.1090/tran/7305 ◽

2017 ◽

Vol 369 (12) ◽

pp. 8319-8342 ◽

Cited By ~ 1

Author(s):

Glen Wheeler ◽

Valentina-Mira Wheeler

Keyword(s):

Free Boundary ◽

Mean Curvature ◽

Mean Curvature Flow ◽

Curvature Flow

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Ancient solutions for Andrews’ hypersurface flow

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2020-0020 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Peng Lu ◽

Jiuru Zhou

Keyword(s):

Ricci Flow ◽

Mean Curvature ◽

Mean Curvature Flow ◽

Curvature Flow ◽

Euclidean Spaces ◽

Round Cylinder ◽

Ancient Solutions ◽

Singular Time

AbstractWe construct the ancient solutions of the hypersurface flows in Euclidean spaces studied by B. Andrews in 1994.As time {t\rightarrow 0^{-}} the solutions collapse to a round point where 0 is the singular time. But as {t\rightarrow-\infty} the solutions become more and more oval. Near the center the appropriately-rescaled pointed Cheeger–Gromov limits are round cylinder solutions {S^{J}\times\mathbb{R}^{n-J}}, {1\leq J\leq n-1}. These results are the analog of the corresponding results in Ricci flow ({J=n-1}) and mean curvature flow.

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