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

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February 2019, 39(2): 863-880. doi: 10.3934/dcds.2019036

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

Kin Ming Hui ^,

Institute of Mathematics, Academia Sinica, Taipei, Taiwan

* Corresponding author: Kin Ming Hui

Received January 2018 Revised August 2018 Published November 2018

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 $\lim_{r\to∞}\frac{rf_r(r)}{f(r)} = \frac{λ (n-1)}{λ (n-1)-1}$.

Keywords: Inverse mean curvature flow, self-similar solution, existence, non-compact solution, asymptotic behaviour.

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

Citation: Kin Ming Hui. Existence of self-similar solutions of the inverse mean curvature flow. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 863-880. doi: 10.3934/dcds.2019036

References:

[1]	B. Allen, Non-compact Solutions to Inverse Mean Curvature Flow in Hyperbolic Space, Ph.D. thesis, University of Tennessee, Knoxville, USA, 2016.
[2]	P. Daskalopoulos and G. Huisken, Inverse mean curvature flow of entire graphs, arXiv: 1709.06665v1
[3]	C. Gerhardt, Flow of nonconvex hypersurfaces into spheres, J. Differential Geom., 32 (1990), 299-314.
[4]	G. Huisken and T. Ilmanen, The Riemannian Penrose inequality, Internat. Math. Res. Notices, 20 (1997), 1045-1058. doi: 10.1155/S1073792897000664.
[5]	G. Huisken and T. Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differential Geom., 59 (2001), 353-437.
[6]	G. Huisken and T. Ilmanen, Higher regularity of the inverse mean curvature flow, J. Differential Geom., 80 (2008), 433-451.
[7]	B. Lambert and J. Scheuer, The inverse mean curvature flow perpendicular to the sphere, Math. Ann., 364 (2016), 1069-1093. doi: 10.1007/s00208-015-1248-2.
[8]	T. Marquardt, Inverse mean curvature flow for star-shaped hypersurfaces evolving in a cone, J. Geom. Anal., 23 (2013), 1303-1313. doi: 10.1007/s12220-011-9288-7.
[9]	T. Marquardt, Weak solutions of the inverse mean curvature flow for hypersurfaces with boundary, J. Reine Angew. Math., 728 (2017), 237-261. doi: 10.1515/crelle-2014-0116.
[10]	K. Smoczyk, Remarks on the inverse mean curvature flow, Asian J. Math., 4 (2000), 331-335. doi: 10.4310/AJM.2000.v4.n2.a3.
[11]	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.

show all references

References:

[1]	B. Allen, Non-compact Solutions to Inverse Mean Curvature Flow in Hyperbolic Space, Ph.D. thesis, University of Tennessee, Knoxville, USA, 2016.
[2]	P. Daskalopoulos and G. Huisken, Inverse mean curvature flow of entire graphs, arXiv: 1709.06665v1
[3]	C. Gerhardt, Flow of nonconvex hypersurfaces into spheres, J. Differential Geom., 32 (1990), 299-314.
[4]	G. Huisken and T. Ilmanen, The Riemannian Penrose inequality, Internat. Math. Res. Notices, 20 (1997), 1045-1058. doi: 10.1155/S1073792897000664.
[5]	G. Huisken and T. Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differential Geom., 59 (2001), 353-437.
[6]	G. Huisken and T. Ilmanen, Higher regularity of the inverse mean curvature flow, J. Differential Geom., 80 (2008), 433-451.
[7]	B. Lambert and J. Scheuer, The inverse mean curvature flow perpendicular to the sphere, Math. Ann., 364 (2016), 1069-1093. doi: 10.1007/s00208-015-1248-2.
[8]	T. Marquardt, Inverse mean curvature flow for star-shaped hypersurfaces evolving in a cone, J. Geom. Anal., 23 (2013), 1303-1313. doi: 10.1007/s12220-011-9288-7.
[9]	T. Marquardt, Weak solutions of the inverse mean curvature flow for hypersurfaces with boundary, J. Reine Angew. Math., 728 (2017), 237-261. doi: 10.1515/crelle-2014-0116.
[10]	K. Smoczyk, Remarks on the inverse mean curvature flow, Asian J. Math., 4 (2000), 331-335. doi: 10.4310/AJM.2000.v4.n2.a3.
[11]	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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