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

Mitake, Hiroyoshi; Zhang, Jinsong

doi:10.1007/s42985-020-00066-4

Cited by 4 publications

(2 citation statements)

References 21 publications

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“…In [31, pages 477-8], he investigated again the Dirichlet problem of the equation studied by Bernstein, changing analyticity of the domain by smoothness as well as he generalized the results to arbitrary dimensions. More recently, equations of type (2) have been considered, specially for the study of radial solutions and the Dirichlet problem: see, for example, [2,3,11,12,13,14,15,26,27].…”

Section: Introductionmentioning

confidence: 99%

Compact surfaces with boundary with prescribed mean curvature depending on the Gauss map

Wang

López

2023

Ann Glob Anal Geom

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Given a C 1 function H defined in the unit sphere S 2 , an H-surface M is a surface in the Euclidean space R 3 whose mean curvature H M satisfies H M (p) = H(N p ), p ∈ M, where N is the Gauss map of M. Given a closed simple curve Γ ⊂ R 3 and a function H, in this paper we investigate the geometry of compact H-surfaces spanning Γ in terms of Γ. Under mild assumptions on H, we prove non-existence of closed H-surfaces, in contrast with the classical case of constant mean curvature. We give conditions on H that ensure that if Γ is a circle, then M is a rotational surface. We also establish the existence of estimates of the area of H-surfaces in terms of the height of the surface.

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Section: Introductionmentioning

confidence: 99%

Compact surfaces with boundary with prescribed mean curvature depending on the Gauss map

Wang

López

2023

Ann Glob Anal Geom

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“…We refer to [21,34,35,36] for detailed analysis on the boundary behavior of such kind of solutions for viscous Hamilton-Jacobi equations. Also, [15,32] study generalized Dirichlet problem for mean curvature flow with a driving force and show large-time convergence to traveling waves with blow-up at boundary. The problem setting in this work is different from these papers: our equation is of nonlinear curvature type and, throughout the evolution, we impose singular Neumann condition directly without prescribing Dirichlet data at all.…”

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confidence: 99%

Singular Neumann boundary problems for a class of fully nonlinear parabolic equations in one dimension

Kagaya¹,

Liu²

2020

Preprint

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In this paper, we discuss singular Neumann boundary problem for a class of nonlinear parabolic equations in one space dimension. Our boundary problem describes motion of a planar curve sliding along the boundary with a zero contact angle, which can be viewed as a limiting model for the capillary phenomenon. We study the uniqueness and existence of solutions by using the viscosity solution theory. Under a convexity assumption on the initial value, we also show the convergence of the solution to a traveling wave as time proceeds to infinity.

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Singular Neumann Boundary Problems for a Class of Fully Nonlinear Parabolic Equations in One Dimension

Kagaya¹,

Liu²

2021

SIAM J. Math. Anal.

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

Cited by 4 publications

References 21 publications

Compact surfaces with boundary with prescribed mean curvature depending on the Gauss map

Compact surfaces with boundary with prescribed mean curvature depending on the Gauss map

Singular Neumann boundary problems for a class of fully nonlinear parabolic equations in one dimension

Singular Neumann Boundary Problems for a Class of Fully Nonlinear Parabolic Equations in One Dimension

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