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

  • * Corresponding author: Kin Ming Hui

    * Corresponding author: Kin Ming Hui
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  • We will give a new proof of a recent result of P. Daskalopoulos, G. Huisken and J.R. King ([2] and reference [7] of [2]) on the existence of self-similar solution of the inverse mean curvature flow which is the graph of a radially symmetric solution in $\mathbb{R}^n$, $n≥ 2$, of the form $u(x,t) = e^{λ t}f(e^{-λ t} x)$ for any constants $λ>\frac{1}{n-1}$ 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 $\text{div}\,\left(\frac{\nabla f}{\sqrt{1+|\nabla f|^2}} \right) = \frac{1}{λ}·\frac{\sqrt{1+|\nabla f|^2}}{x·\nabla f-f}$ in $\mathbb{R}^n$, $f(0) = μ$, for any $λ>\frac{1}{n-1}$ 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 $\lim_{r\to∞}\frac{rf_r(r)}{f(r)} = \frac{λ (n-1)}{λ (n-1)-1}$.

    Mathematics Subject Classification: Primary: 35K67, 35J75; Secondary: 53C44.

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  •   B. Allen, Non-compact Solutions to Inverse Mean Curvature Flow in Hyperbolic Space, Ph.D. thesis, University of Tennessee, Knoxville, USA, 2016.
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      J. Urbas , On the expansion of starshaped hypersurfaces by symmetric functions of their principle curvatures, Math. Z., 205 (1990) , 355-372.  doi: 10.1007/BF02571249.
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