Linearized stability analysis of surface diffusion for hypersurfaces with triple lines

Depner, Daniel; Garcke, Harald

doi:10.14492/hokmj/1362406637

Cited by 14 publications

(40 citation statements)

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“…Theorem 1 of this paper generalises these results to arbitrary codimension and to the case of surfaces with boundary. The case of anisotropic surface diffusion flow has recently received some attention [5,6], where one studies the steepest descent H −1 -gradient flow of the functional Σ a dµ. Stationary solitons for this flow are also covered by our theorems.…”

Section: Setting and Main Resultsmentioning

confidence: 97%

Gap phenomena for a class of fourth-order geometric differential operators on surfaces with boundary

Wheeler

2014

Proc. Amer. Math. Soc.

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Abstract. In this paper we establish a gap phenomenon for immersed surfaces with arbitrary codimension, topology and boundaries that satisfy one of a family of systems of fourth-order anisotropic geometric partial differential equations. Examples include Willmore surfaces, stationary solitons for the surface diffusion flow, and biharmonic immersed surfaces in the sense of Chen. On the boundary we enforce either umbilic or flat boundary conditions: that the tracefree second fundamental form and its derivative or the full second fundamental form and its derivative vanish. For the umbilic boundary condition we prove that any surface with small L 2 -norm of the tracefree second fundamental form or full second fundamental form must be totally umbilic; that is, a union of pieces of round spheres and flat planes. We prove that the stricter smallness condition allows consideration for a broader range of differential operators. For the flat boundary condition we prove the same result with weaker hypotheses, allowing more general operators, and a stronger conclusion: only pieces of planes are allowed. The method used relies only on the smallness assumption and thus holds without requiring the imposition of additional symmetries. The result holds in the class of surfaces with any genus and irrespective of the number or shape of the boundaries.

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Section: Setting and Main Resultsmentioning

confidence: 97%

Gap phenomena for a class of fourth-order geometric differential operators on surfaces with boundary

Wheeler

2014

Proc. Amer. Math. Soc.

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“…For the linearization of (4.20) around ρ ≡ 0, that is around the stationary solution Γ * , we refer to [9] (see also [10]). More precisely, the linearization of the surface diffusion equation is done in [9, Lemma 3.2] and a similar argument as in [9,Lemma 3.4] gives the following linearization of the angle condition ∂ n ∂Γ * ρ + κ n ∂Γ * µ = 0 on ∂Γ * .…”

Section: Linearization and General Settingmentioning

confidence: 99%

On convergence of solutions to equilibria for fully nonlinear parabolic systems with nonlinear boundary conditions

2015

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Convergence to stationary solutions in fully nonlinear parabolic systems with general nonlinear boundary conditions is shown in situations where the set of stationary solutions creates a C 2 -manifold of finite dimension which is normally stable. We apply the parabolic Hölder setting which allows to deal with nonlocal terms including highest order point evaluation. In this direction some theorems concerning the linearized systems is also extended. As an application of our main result we prove that the lens-shaped networks generated by circular arcs are stable under the surface diffusion flow.

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“…Linearization of the mean curvature: For the linearization of the mean curvature H i , we use the following result, see Depner, Garcke [8] and Depner [7], where [7] contains the detailed calculation:…”

Section: Linearization Of the Normal Velocity: For The Linearization mentioning

confidence: 99%

“…Linearization of the angle conditions: The linearization of the angle condition N i , N j = cos θ k is the technically most challenging part and we use the following result of Depner and Garcke [8]:…”

Section: Linearization Of the Normal Velocity: For The Linearization mentioning

confidence: 99%

Mean Curvature Flow with Triple Junctions in Higher Space Dimensions

Depner

Garcke

Kohsaka

2013

Arch Rational Mech Anal

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We consider mean curvature flow of n-dimensional surface clusters. At (n − 1)-dimensional triple junctions an angle condition is required which in the symmetric case reduces to the well-known 120 degree angle condition. Using a novel parametrization of evolving surface clusters and a new existence and regularity approach for parabolic equations on surface clusters we show local well-posedness by a contraction argument in parabolic Hölder spaces.

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Linearized stability analysis of surface diffusion for hypersurfaces with triple lines

Cited by 14 publications

References 0 publications

Gap phenomena for a class of fourth-order geometric differential operators on surfaces with boundary

Gap phenomena for a class of fourth-order geometric differential operators on surfaces with boundary

On convergence of solutions to equilibria for fully nonlinear parabolic systems with nonlinear boundary conditions

Mean Curvature Flow with Triple Junctions in Higher Space Dimensions

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