By G Lefort
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Additional info for Algebra & Analysis, Problems & Solutions
The reason for introducing the additional term ηH will be explained later. A straightforward calculation yields the evolution equation 2(1 − σ) σ(1 − σ) ∂f = Δf + ∇H, ∇f − f |∇H|2 ∂t H H2 ∂ 2 Qk+1 m i 1 + 1−σ ∇ hj ∇m hpq + σ|A|2 f. 1) The presence of the reaction term due to σ does not allow us to estimate f from above directly using the maximum principle. Therefore, we use a strategy similar to the one of . The crucial step consists of deriving Lp estimates on the positive part of f for large p.
6. Let M0 be the graph of a function xn+1 = u(x1 , . . , xn ) deﬁned for all (x1 , . . , xn ) ∈ IRn . Then there exists a solution Mt of the mean curvature ﬂow with initial data M0 which is deﬁned for all t ∈ (0, +∞) and is a graph over IRn for all t. , the linear heat equation), no growth restriction on the function u is necessary to obtain existence of solutions. This is related to the geometric interpretation of the equation: in fact, a fast growth of u does not imply a fast growth of the curvature of the graph of u.
Then, any ﬂow Mτ , with τ ∈ (−∞, Ω), obtained as a limit of rescalings by the procedure of the previous section, is a homothetically shrinking solution of the mean curvature ﬂow. 7) where M0 is the image of an immersion F0 which satisﬁes F0 · ν = 2Ω H. 8) Proof. We only give a sketch of the proof of this result, which can be found in  and is based on the monotonicity formula. Let us denote by F : M × (−∞, Ω) → IRn+1 the immersion obtained as a limit of rescalings such that F (M, τ ) = Mτ . 1), with t¯ = Ω, must be identically zero on Mτ .