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A NONLINEAR PARTIAL

A NONLINEAR PARTIAL DIFFERENTIAL EQUATION FOR THE VOLUME PRESERVING MEAN CURVATURE FLOW3 the identity function on Γ. Moreover, S is said to be in the class H s (Γ) if ϱ is in the class H s . (,t) S t 1 0 Figure 1: Normal graph over the unit sphere in R 2 . Let Γ be the unit sphere in R n+1 of zero center and consider a family {S t , t ≥ 0} of closed hypersurfaces in R n+1 where for any t ≥ 0 S t is a graph in the normal direction over Γ (see Fig. 1). More specifically, for Γ = {x ∈ R n+1 : |x| = 1}, we assume that there exists function ρ ∗ : Γ × R → R defining for t fixed a diffeomorphism θ ρ ∗(·, t) onto S t : θ ρ ∗ : Γ × t → S t : θ ρ ∗(γ, t) := γ + ρ ∗ (γ, t)ν(γ), γ ∈ Γ, t ≥ 0. Since θ ρ ∗(Γ, t) = S t we deduce that in cartesian coordinates x 1 , · · · , x n+1 , S t is represented by (2.1) S t = where R n+1 ∗ { x ∈ R n+1 ∗ : |x| − 1 − ρ ∗( x ) } |x| , t = 0 , := R n+1 −{0}. By setting ̂ρ := 1+ρ ∗ , [2], we define the diffeomorphism θ̂ρ (γ, t) := ̂ρ(γ, t)γ = θ ρ ∗(γ, t) − id(γ). We represent S t by using the diffeomorphism θ̂ρ . S t is identified by the function ̂ρ(·, t) : Γ → R. Let x = (x 1 , · · · , x n+1 ) ∈ R n+1 in cartesian coordinates and consider the change of variables in polar coordinates u = (u 1 , · · · , u n+1 ) (2.2) x = x(u) = ( ) x 1 (u 1 , · · · , u n+1 ), · · · , x n+1 (u 1 , · · · , u n+1 ) , x 1 = u n+1 y 1 (u 1 , · · · , u n ), · · · , x n+1 = u n+1 y n+1 (u 1 , · · · , u n ), where u n+1 = |x| ∈ [0, +∞). The function y = (y 1 , · · · , y n+1 ) is on Γ and y i i = 1, · · · , n+1 in polar coordinates may be expressed by the following formulas for n = 1, 2, ([10]) (2.3) (n = 1) y 1 = cos(u 1 ), y 2 = sin(u 1 ), (n = 2) y 1 = cos(u 1 ) cos(u 2 ), y 2 = sin(u 1 ) cos(u 2 ), y 3 = sin(u 2 ),

4 DIMITRA ANTONOPOULOU, GEORGIA KARALI where u 1 , u 2 ∈ [0, 2π] × [0, π). If n ≥ 3 then y can be defined by (2.4) y 1 = cos(u 1 ), y k = y n = ( n−1 ∏ j=1 ( k−1 ∏ j=1 ) sin(u j ) cos(u n ), y n+1 = ) sin(u j ) cos(u k ), k = 2, · · · , n − 1, ( n−1 ∏ j=1 ) sin(u j ) sin(u n ), ([11]), where 0 ≤ u j < π for j = 1, · · · , n − 1, 0 ≤ u n < 2π. Note that the geometrical properties of S t are independent of the choice of parametrization y of Γ. We may write u = u(x) = (u 1 (x 1 , · · · , x n+1 ), · · · , u n+1 (x 1 , · · · , x n+1 )) as the change of variables is invertible. ( ) Remark 2.1. Obviously, ρ ∗ x |x| , t is a function of x = (x 1 , · · · , x n+1 ) and t for any x ∈ R n+1 ∗ , therefore, for t fixed we define by ˜ρ(·, t) : R n+1 ∗ → R ˜ρ(x 1 , · · · , x n+1 , t) := 1 + ρ ∗( x ) ( x ) |x| , t =: ̂ρ |x| , t . Denote that the above gives that ˜ρ is independent from |x| while it depends only on the directional angles, and thus any change of variables of R n+1 from cartesians to polar coordinates will give for x ∈ R n+1 ∗ ˜ρ(x 1 , · · · , x n+1 , t) = ρ(u 1 , · · · , u n , u n+1 , t) = ρ(u 1 , · · · , u n , t), since u n+1 := |x|. In this paper we compute ρ as a solution of an initial and boundary value problem. Then we may use this ρ to construct S t as follows: If Γ is represented by Γ := {y ∈ R n+1 : y = (y 1 (u 1 , · · · , u n ), · · · , y n+1 (u 1 , · · · , u n ))}, where y is given for example by (2.3), (2.4) then (2.5) { ( )} S t := x ∈ R n+1 : x = ρ(u 1 , · · · , u n , t) y 1 (u 1 , · · · , u n ), · · · , y n+1 (u 1 , · · · , u n ) . 3. The evolution equation In this Section we consider S t to be a normal graph over the unit sphere Γ defined by (2.1) and prove an equivalent formulation for (1.1) presented as an evolution equation in time for ρ = ρ(u 1 , · · · , u n , t) in polar coordinates. Then S t may be constructed in R n+1 by utilizing (2.5). We prove the next lemma. Lemma 3.1. If S t satisfies the VPMCF (1.1) then ˜ρ satisfies (3.1) ∂ t˜ρ = h √ 1 + |∇ x˜ρ| 2 + 1 { − n n |x| +∆ |∇ x˜ρ| 2 x˜ρ− |x|(1 + |∇ x˜ρ| 2 ) −∇ x˜ρ Hess x (˜ρ) ∇ x˜ρ T }∣ ∣∣x∈St (1 + |∇ x˜ρ| 2 . )

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