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A NONLINEAR PARTIAL

A NONLINEAR PARTIAL DIFFERENTIAL EQUATION FOR THE VOLUME PRESERVING MEAN CURVATURE FLOW7 and consequently (3.14) n+1 ∑ j=1 x j |x| ( n+1 ∑ i=1 ) ˜ρ xi ˜ρ xix j = − 1 n+1 ∑ ˜ρ xi ˜ρ xi = − |∇ x˜ρ| 2 . |x| |x| i=1 By (3.13) combined with (3.14) the next relation follows B = − |∇ x˜ρ| 2 |x| − ∇ x˜ρ Hess x (˜ρ) ∇ x˜ρ T . We replace B and A in (3.11) to arrive at (3.15) n nH = |x| √ 1 + |∇ − ∆ √ x˜ρ + |∇ x˜ρ| 2 x˜ρ| 2 1 + |∇x˜ρ| 2 |x|(1 + |∇ x˜ρ| 2 ) 3 2 Plugging the above in (3.9) we obtain equation (3.1). + ∇ x˜ρ Hess x (˜ρ) ∇ x˜ρ T (1 + |∇ x˜ρ| 2 ) 3 2 ∣ x∈St . The non-linear evolution equation for the VPMCF in polar coordinates is presented in the next theorem. Theorem 3.2. Let S t be a graph in normal direction over Γ determined by the function ρ(·, t) : Γ → R. If S t satisfies the VPMCF (1.1) then ρ satisfies the evolution equation (3.16) ∂ t ρ = G(ρ), t ≥ 0, where Here J(ρ) is defined as (3.17) J(ρ) := 1 n with [ ∑ n n∑ (3.18) R 1 (ρ) := G(ρ) := J(ρ) + h ρ √ ρ2 + R 2 (ρ). { − n ρ + R 1(ρ) ρ 2 − i=1 j=1 (3.19) R 2 (ρ) := (3.20) R 3 (ρ) := ∑ ∑ ([ ∑ n n+1 n+1 j=1 i=1 [ n ∑ n∑ l=1 k=1 s=1 ∂ 2 n+1 ρ ∑ ∂u i ∂u j n+1 ∑ n∑ R 2 (ρ) ρ(ρ 2 + R 2 (ρ)) − m=1 n∑ k=1 m=1 i=1 ∂ρ ∂u ] s ∂u s ∂y i ∂ 2 ρ ∂u l ∂u k ∂u k ∂y i ∂u l ∂y j + ∂u j ∂y m ∂u i ∂y m + ∂ρ ∂u m ∂u m ∂y k n∑ ∂ρ ∂u q q=1 R 3 (ρ) } ρ 2 (ρ 2 , + R 2 (ρ)) n∑ q=1 ∂ρ ∂u i ∂u i ∂y k , n+1 ∂ρ ∑ ∂u q m=1 ∂ 2 u ][ q ∑ n ∂y i ∂y j while ∫ S h := t H dσ ( ∫ )( ∫ ) −1, ∫ S t dσ = − ρJ(ρ)(ρ 2 + R 2 (ρ)) − 1 2 µρ µ ρ Γ Γ where µ ρ is the Jacobian in polar coordinates. q=1 ∂ 2 u ] q ∂ym 2 , ∂ρ ∂u q ∂u q ∂y j ]) , □

8 DIMITRA ANTONOPOULOU, GEORGIA KARALI Proof. By Lemma 3.1 the VPMCF (1.1) is transformed to (3.1). We express (3.1) in terms of ρ. First, we calculate ∆ x˜ρ(x)| x∈St . Let x ∈ R n+1 ∗ , then for ˜ρ(x, t) = ρ(u 1 , · · · , u n , t) (polar coordinates in space), we apply the chain rule and use that x = u n+1 y = |x|y. In details, for any x ∈ R n+1 ∗ , we consider and compute (3.21) ˜ρ xl x m = n∑ n∑ i=1 j=1 = 1 |x| 2 [ n ∑ ˜ρ(x, t) = ρ(u 1 , · · · , u n , t) ∂ 2 ρ ∂u i ∂u j ∂u j ∂x l n∑ i=1 j=1 for any l, m ≤ n + 1. Here we used that and that ∂u i ∂x m + ∂ 2 ρ ∂u i ∂u j ∂u j ∂y l ∂u i = 1 ∂x m |x| ∂u i , ∂y m n∑ ∂ρ ∂ 2 u q ∂u q ∂x l ∂x m q=1 ∂u i ∂y m + ∂ 2 u q = 1 ∂ 2 u q ∂x l ∂x m |x| 2 , ∂y l ∂y m n∑ ∂ρ ∂ 2 u ] q , ∂u q ∂y l ∂y m since for any x ∈ R n+1 ∗ there exists y ∈ Γ such that x = |x|y. Note that y is defined as a parametrization of Γ, for example by (2.3), (2.4). Hence we obtain (3.22) ∆ x˜ρ = = n+1 ∑ m=1 ˜ρ xmx m = 1 n+1 ∑ |x| 2 1 [ ∑ n |x| 2 n∑ i=1 j=1 m=1 [ n ∑ ∂ 2 n+1 ρ ∑ ∂u i ∂u j m=1 n∑ i=1 j=1 q=1 ∂ 2 ρ ∂u i ∂u j ∂u j ∂y m ∂u i ∂y m + ∂u j ∂y m ∂u i ∂y m + n∑ q=1 n+1 ∂ρ ∑ ∂u q m=1 n∑ q=1 ∂ 2 u q ∂y 2 m ∂ρ ∂ 2 u ] q ∂u q ∂ym 2 If x ∈ S t then (3.2), (3.3) give that |x| = 1 + ρ ∗ ( x |x| , t) = ρ(u 1, · · · , u n , t). Thus (3.22) yields (3.23) ∆ x˜ρ(x)| x∈St = 1 ρ 2 R 1(ρ), for R 1 (ρ) defined by (3.18). We note that the terms appearing in (3.18) can be computed because y is a known function given for example by (2.3) or (2.4). Applying the chain rule at ˜ρ(x, t) = ρ(u 1 , · · · , u n , t) we arrive at for any k ≤ n + 1, consequently ∂˜ρ ∂x k = k=1 n∑ l=1 ∂ρ ∂u l ∂u l ∂x k , n+1 ∑ {( ∑ n (3.24) |∇ x˜ρ| 2 ∂ρ ∂u ) 2 } l = . ∂u l ∂x k l=1 By (3.24) and by making use of the identity ( ∑ n ) 2 ε l = l=1 n ∑ m=1 i=1 n∑ ε m ε i ] .

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