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A NONLINEAR PARTIAL

A NONLINEAR PARTIAL DIFFERENTIAL EQUATION FOR THE VOLUME PRESERVING MEAN CURVATURE FLOW11 on the azimuthal derivatives for any u n ∈ [0, 2π] and any t > 0 as follows ∂ρ (0, u 2 , · · · , u n , t) = 0, ∂ un ∂ρ (u 1 , 0, · · · , u n , t) = 0, ∂ un ............................ ∂ρ (u 1 , u 2 , · · · , 0, u n , t) = 0, ∂ un u i ∈ (0, π), i ≠ n − 1, n. ∂ρ (π, u 2 , · · · , u n , t) = 0, u i ∈ (0, π), i ≠ 1, n, ∂ un ∂ρ (u 1 , π, · · · , u n , t) = 0, u i ∈ (0, π), i ≠ 2, n, ∂ un ∂ρ ∂ un (u 1 , u 2 , · · · , π, u n , t) = 0, For the case n = 1 we only consider periodicity on azimuth. Remark 3.6. For the 2-dimensional VPMCF (3.16), i.e. in the case of curves in R 2 , let n = 1. We use the symbol u 1 =: θ, to obtain ρ = ρ(θ, t), y 1 = cos θ, y 2 = sin θ, θ = arctan( y2 y 1 ), 0 ≤ θ ≤ 2π, and we further compute ∂θ ∂θ ∂y 1 = − sin θ, ∂y 2 = cos θ, ∂ 2 θ ∂ = −2 cos θ sin θ, 2 θ ∂y1 2 ∂y 1∂y 2 = −1 + 2 cos 2 θ, ∂2 θ = 2 cos θ sin θ. We replace in R ∂y2 2 1 , R 2 , R 3 e.g. (3.18)-(3.20), to obtain after straightforward calculations that Thus by (3.17) we get R 1 (ρ) = ρ θθ , R 2 (ρ) = ρ 2 θ, R 3 (ρ) = ρ 2 θρ θθ . J(ρ) = ρ θθ − ρ2 θ ρ ρ 2 + ρ 2 − 1 θ ρ . In order to calculate h we write { } S t = z ∈ R 2 : z = (z 1 (θ, t), z 2 (θ, t)) = ρ(θ, t)(cos θ, sin θ), θ ∈ [0, 2π] , thus ∫ S t dσ = ∫ 2π √ 0 z 2 1θ + z2θ 2 dθ. We compute z2 1θ + z2 2θ = ρ2 θ + ρ2 to get ∫ S t dσ = ∫ 2π 0 √ ρ 2 θ + ρ2 dθ, h = − We replace in (3.16) and obtain the final equation (3.28) ∂ t ρ = ρ θθ − ρ2 θ ρ ρ 2 + ρ 2 θ ∫ 2π 0 ∫ 2π ∫ 2π ρJ(ρ)dθ 0 √ ρ 2 θ + ρ 2 dθ . − 1 √ ρ − ρ2 + ρ 2 θ ρJ(ρ)dθ 0 ρ ∫ 2π √ . 0 ρ 2 θ + ρ 2 dθ Remark 3.7. In three dimensions (n = 2) using (2.3) we write { } S t = z ∈ R 3 : z = ρ(θ, φ, t)(cos θ cos φ, sin θ cos φ, sin φ), θ ∈ [0, 2π], φ ∈ [0, π] . In this case we may compute H by using the first and second fundamental forms for hypersurfaces. 4. Numerical experiments for the 2-dimensional VPMCF 4.1. Finite difference schemes. We consider the case n = 1. The VPMCV can be presented (Remark 3.6, eqn. (3.28)) as the following non-linear initial and

12 DIMITRA ANTONOPOULOU, GEORGIA KARALI boundary value problem for ρ ∂ t ρ = ρ θθ − ρ2 θ ρ ρ 2 + ρ 2 θ − 1 ρ + √ρ 2 + ρ 2 θ ∫ ( 2π −ρθθ + ρ2 θ ) ρ + 1 0 ρ 2 +ρ 2 ρ ρ dθ θ ρ ∫ 2π √ ρ2 + ρ 2 0 θ dθ , 0 < θ < 2π, t > 0, (4.1)ρ(θ, 0) = ρ 0 (θ), 0 ≤ θ ≤ 2π, ρ(0, t) = ρ(2π, t), t ≥ 0, with periodic conditions and smooth periodic initial data ρ 0 . We will consider the case when ρ 0 is non-convex. We approximate numerically (4.1) by explicit finite difference schemes using the trapezoid rule for the non-local integral terms. More specifically, let define the uniform partition 0 = θ 0 < θ 1 < θ 2 < · · · < θ J = 2π, with θ j := jh, j = 0, · · · J, for h := 2π 100 , J := 100. We approximate the terms for j = 1, · · · , J − 1, by ρ n j , ρ(θ j , t n ), ∂ t ρ(θ j , t n+1 ), ρ θ (θ j , t n ), ρ θθ (θ j , t n ) ρ n+1 j − ρ n j , k ρ n j+1 − ρn j−1 2h and ρn j+1 − 2ρn j + ρn j−1 h 2 respectively, for k = 1 100 , and tn := nk, n = 0, · · · , N. We also approximate the values ρ(θ 0 , t n+1 ) = ρ(θ J , t n+1 ) by ρ n+1 1 for n = 0, · · · , N, and use the initial condition ρ 0 j := ρ 0(θ j ), j = 1, · · · , J. Obviously while for any t > 0 S 0 = {z ∈ R 2 : z = ρ 0 (θ)(cos θ, sin θ), θ ∈ [0, 2π]}, S t = {z ∈ R 2 : z = ρ(θ, t)(cos θ, sin θ), θ ∈ [0, 2π]}. In details, let ρ 0 j := ρ 0(θ j ), j = 1, · · · , J. For any n = 1, · · · , N we solve the J − 1 × J − 1 diagonal system ( ) ρ n 2 ρ n+1 j − ρ n ρ n j+1 −ρn √ j−1 ( ) j+1 −2ρn j +ρn 2h j−1 j h = 2 − ρ n j ( ) 2 − 1 (ρ n ρ n 2 j )2 j+1 + −ρn j−1 2h A k ρ + n n j ρ n j B n , (ρ n j )2 + ρ n j+1 −ρn j−1 2h where j = 1, · · · , J − 1, while A n , B n are the approximations of the non-local integral terms of (4.1) at t := t n calculated by the trapezoid rule. Further using the periodic conditions for j = J and any n = 0, · · · , N we set ρ n+1 0 := ρ n+1 1 , ρ n+1 J := ρ n+1 1 . 4.2. Numerical results. For the first experiment (Case 1), we use as initial condition the following non-convex smooth and periodic function ρ 0 (θ) = (4 + cos 3 θ)(2 + sin 3 θ). Figure 2 presents the evolution of the closed initial curve S 0 for various times t 0 = 0, t 1 = 1, t 2 = 10, t 3 = 20. In this case the asymptotic convergence to a sphere is observable.

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