Existence of self-similar solution of the inverse mean curvature flow

2018/01/25 by K. M. Hui, Hui, K. M.
Mathematics · #35J75 Secondary 53C44 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Primary 35K67 #math.AP #math.DG #msc:35J75 #msc:35K67 #msc:53C44

paper · pdf · doi:10.48550/arxiv.1801.08250

21 pages, proof of Theorem 1.2 completely rewritten

arxiv created 2018/08/23 · arxiv updated 2018/08/24

Abstract

We will give a new proof of a recent result of P.~Daskalopoulos, G.Huisken and J.R.King ([DH] and reference [7] of [DH]) on the existence of self-similar solution of the inverse mean curvature flow which is the graph of a radially symmetric solution in ℝⁿ, n≥ 2, of the form u(x,t)=e^λtf(e^-λt x) for any constants λ>(1)/(n-1) and μ<0 such that f(0)=μ. More precisely we will give a new proof of the existence of a unique radially symmetric solution f of the equation div ((∇ f)/(√(1+|∇ f|²)) )=\frac1λ⋅(√(1+|∇ f|²))/(x⋅∇ f-f) in ℝⁿ, f(0)=μ, for any λ>(1)/(n-1) and μ<0, which satisfies f_r(r)>0, f_rr(r)>0 and rf_r(r)>f(r) for all r>0. We will also prove that lim_r→∞(rf_r(r))/(f(r))=(λ(n-1))/(λ(n-1)-1).

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