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A NONLINEAR PARTIAL

A NONLINEAR PARTIAL DIFFERENTIAL EQUATION FOR THE VOLUME PRESERVING MEAN CURVATURE FLOW9 for we obtain Thus we get |∇ x˜ρ| 2 = n+1 ∑ n∑ k=1 m=1 i=1 = 1 n+1 ∑ |x| 2 ε l := ∂ρ ∂u l ∂u l ∂x k , n∑ n∑ k=1 m=1 i=1 ∂ρ ∂u m ∂u m ∂x k n∑ ∂ρ ∂u i ∂u i ∂x k ∂ρ ∂u m ∂u m ∂y k (3.25) |∇ x˜ρ| 2 | x∈St = 1 ρ 2 R 2(ρ), ∂ρ ∂u i ∂u i ∂y k , x ∈ R n+1 . for R 2 (ρ) defined by (3.19). Next, we express in terms of ρ the operator ∇ x˜ρ Hess x (˜ρ) ∇ x˜ρ T appearing in (3.1). The equality ˜ρ(x 1 , · · · , x n+1 , t) = ρ(u 1 , · · · , u n , t) yields n∑ ∂ρ ∂u l n∑ n∑ ∂ 2 ρ ∂u k ∂u l n∑ ∂ρ ∂ 2 u q ˜ρ xj = , ˜ρ xix ∂u l ∂x j = + . j ∂u l ∂u k ∂x i ∂x j ∂u q ∂x i ∂x j l=1 l=1 k=1 By (3.12) and the above, we obtain for any x ∈ R n+1 (3.26) n+1 ∑ n+1 ∑ ∇ x˜ρ Hess x (˜ρ) ∇ x˜ρ T = ˜ρ xi ˜ρ xix j ˜ρ xj = ∑ ∑ ([ ∑ n n+1 n+1 j=1 i=1 [ n ∑ q=1 [ n ∑ s=1 j=1 i=1 ∂ρ ∂u s ∂u s ∂x i ][ n ∑ n∑ l=1 k=1 ∂ρ ∂u ]) q = 1 n+1 ∑ n+1 ∑ ∂u q ∂x j |x| 4 n∑ l=1 k=1 Hence, we compute j=1 i=1 ∂ 2 ρ ∂u l ∂u k ∂u k ∂y i ∂u l ∂y j + q=1 ∂ 2 ρ ∂u l ∂u k ∂u k ∂x i ∂u l ∂x j + ([ n ∑ n∑ ∂ρ ∂u q q=1 s=1 ∂ρ ∂u ] s ∂u s ∂y i ∂ 2 u ][ q ∑ n ∂y i ∂y j q=1 (3.27) ∇ x˜ρ Hess x (˜ρ) ∇ x˜ρ T | x∈St = 1 ρ 4 R 3(ρ), n∑ ∂ρ ∂ 2 u ] q ∂u q ∂x i ∂x j q=1 ∂ρ ∂u q ∂u q ∂y j ]) . for R 3 (ρ) given by (3.20). Utilizing the definition of J(ρ) in (3.17), the expression of H in (3.15) and the definitions of R 1 (ρ), R 2 (ρ), R 3 (ρ), we obtain that hence (3.25) yields J(ρ) = −H √ 1 + |∇ x˜ρ| 2 , H = −J(ρ)ρ √ ρ2 + R 2 (ρ) . Using the previous expression in h = ∫ H dσ S t dσ ∫S t and the values given by (3.23), (3.25), (3.27) in (3.1) we get finally (3.16) since ∂ t˜ρ = ∂ t ρ. □

10 DIMITRA ANTONOPOULOU, GEORGIA KARALI Remark 3.3. The operators R 1 , R 2 and R 3 , e.g. (3.18)-(3.20), may be expressed in terms of the Beltrami differential parameters of first and second order. Considering the first fundamental form G = (G) ij , i, j = 1, · · · , n of the surface ρ we define in cartesians where ∂ k ρ := ∂ρ := (∂ 1 ρ, · · · , ∂ n ρ), n∑ m=1 g km while g km are the elements of G −1 . Further let ∂ρ , k = 1, · · · , n, ∂u m |∂ρ| 2 G := ∂ρG∂ρ T . This expression is equal to the first differential parameter of Beltrami which is invariant with respect to allowable transformations of coordinates, [10], [2]. We also define ∆ Γ ρ as the Laplace-Beltrami operator on the unit sphere which is invariant too and also called as second differential parameter of Beltrami, [10]. Finally, let the n × n matrix Hessρ be for t fixed the second covariant derivative of the scalar function ρ(u 1 , · · · , u n , t), [10], given by (Hessρ) rs := ∂2 ρ ∂u r ∂u s − n∑ p=1 Γ p rs ∂ρ , ∂u p where Γ p rs are the Christoffel symbols of second kind, then it follows that R 1 (ρ) = ∆ Γ ρ, R 2 (ρ) = |∂ρ| 2 G, R 3 (ρ) = ∂ρ Hessρ ∂ρ T , and the VPMCF (1.1) admits an elegant representation in terms of Beltrami operators and of covariant Hessian, [2]. Remark 3.4. Considering the standard parametrization (2.4), we denote that ( ) u i = arcot y i (yi 2 + yi+1 2 + · · · + yn+1) 2 − 1 2 , i = 1, · · · , n. Remark 3.5. In order to compute an explicit formula for (3.16), we may use the standard parametrization given by (2.3), or (2.4). We supplement the non-linear p.d.e. (3.16) by an initial periodic condition ρ(·, 0) given for any (u 1 , · · · , u n ) ∈ (0, π) × · · · × (0, π) × [0, 2π]. We also impose periodic and Dirichlet boundary conditions and derive an initial and boundary value problem. More specifically, if n ≥ 2 we consider the p.d.e. (3.16) at any t > 0 for (u 1 , · · · , u n−1 ) in the open set A := (0, π) × · · · × (0, π), and for any u n ∈ (0, 2π). Since S t is closed we impose a periodic boundary condition on the azimuth u n by ρ(u 1 , · · · , u n−1 , 0, t) = ρ(u 1 , · · · , u n−1 , 2π, t) for any (u 1 , · · · , u n−1 ) ∈ A, t ≥ 0. In addition, we assign boundary values at the south and north poles 0, π. Since the coordinate system is polar, these values at the poles must be independent of the azimuth u n (u n is measured along the equator), so we impose Dirichlet conditions

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