Stability Analysis of Delaunay Surfaces as Steady States for the Surface Diffusion Equation
The stability of steady states for the surface diffusion equation will be studied. In the axisymmetric setting, steady states are the Delaunay surfaces, which are the axisymmetric constant mean curvature surfaces. We consider a linearized stability of the
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Abstract The stability of steady states for the surface diffusion equation will be studied. In the axisymmetric setting, steady states are the Delaunay surfaces, which are the axisymmetric constant mean curvature surfaces. We consider a linearized stability of these surfaces and derive criteria of the stability by investigating the sign of eigenvalues corresponding to the linearized problem. Keywords Surface diffusion equation · Delaunay surfaces · Stability
1 Introduction Let Γt ⊂ R3 be a moving surface with respect to time t governed by the geometric evolution law (1) V = −ΔΓt H on Γt , where V is the normal velocity of Γt , H is the mean curvature of Γt , and ΔΓt is the Laplace-Beltrami operator on Γt . In our sign convention, the mean curvature H for spheres with outer unit normal is negative. Equation (1) is called surface diffusion equation. The surface diffusion equation (1) was first derived by Mullins [12] to model the motion of interfaces in the case that the motion of interfaces is governed purely by mass diffusion within the interfaces. (For simplicity, we choose 1 as the diffusion constant.) Also, Cahn and Taylor [14] showed that (1) is the H −1 -gradient flow of the area functional of Γt , so that this geometric evolution equation has a variational structure that the area of the surface decreases whereas the volume of the region enclosed by the surface is preserved.
Y. Kohsaka (B) Graduate School of Maritime Sciences, Kobe University, 5-1-1, Fukae-minamimachi, Higashinada-ku, Kobe 658-0022, Japan e-mail: [email protected] © Springer International Publishing Switzerland 2016 F. Gazzola et al. (eds.), Geometric Properties for Parabolic and Elliptic PDE’s, Springer Proceedings in Mathematics & Statistics 176, DOI 10.1007/978-3-319-41538-3_8
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Y. Kohsaka Π−
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Fig. 1 Setting of (2)
In this paper, we consider the following problem. For φ± : R+ → R, set Π± = {(φ± (|η|), η)T | η ∈ R2 }, Ω = {(x, η)T | φ− (|η|) ≤ x ≤ φ+ (|η|), η ∈ R2 }. Note that ∂Ω = Π− or Π+ . Let us assume that Γt ⊂ Ω and the motion of Γt is governed by ⎧ V = −ΔΓt H on Γt , ⎪ ⎪ ⎨ (NΓt , NΠ± )R3 = cos θ± on Γt ∩ Π± , (2) (∇Γt H, ν± )R3 = 0 on Γt ∩ Π± , ⎪ ⎪ ⎩ Γt |t=0 = Γ0 . Here, NΓt and NΠ± are the outer unit normals to Γt and Π± (= ∂Ω), respectively, and ν± are the outer unit co-normals to ∂Γt on Γt ∩ Π± . The problem (2) are obtained as the H −1 -gradient flow of the capillary energy Area [Γt ] + μ+ Area [Σt,+ ] + μ− Area [Σt,− ], where Σt,± are the part of Π± with the boundary ∂Σt,± = Γt ∩ Π± . Note that contact angles θ± are given by cos θ± = μ± (Fig. 1). Let Γ∗ be the steady states for (2) and H∗ be the mean curvature of Γ∗ . Then H∗ satisfies ΔΓ∗ H∗ = 0 on Γ∗ , (∇Γ∗ H∗ , ν± )R3 = 0 on Γ∗ ∩ Π± . This implies ∇Γ∗ H∗ 2L 2 (Γ∗ ) = 0, so that we see that the steady states of (2) are the constant mean curvature surfaces (CMC surfaces). In this paper, we consider the axisymmetric CMC surfaces, which are called Delaunay surfaces, as the steady states Γ∗ , and analyze the eigenvaule problem
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