American Institute of Mathematical Sciences

Advanced
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 & Continuous Dynamical Systems - A, 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. Google Scholar

[2]

P. Daskalopoulos and G. Huisken, Inverse mean curvature flow of entire graphs, arXiv: 1709.06665v1 Google Scholar

[3]

C. Gerhardt, Flow of nonconvex hypersurfaces into spheres, J. Differential Geom., 32 (1990), 299-314.   Google Scholar

[4]

G. Huisken and T. Ilmanen, The Riemannian Penrose inequality, Internat. Math. Res. Notices, 20 (1997), 1045-1058.  doi: 10.1155/S1073792897000664.  Google Scholar

[5]

G. Huisken and T. Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differential Geom., 59 (2001), 353-437.   Google Scholar

[6]

G. Huisken and T. Ilmanen, Higher regularity of the inverse mean curvature flow, J. Differential Geom., 80 (2008), 433-451.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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. Google Scholar

[2]

P. Daskalopoulos and G. Huisken, Inverse mean curvature flow of entire graphs, arXiv: 1709.06665v1 Google Scholar

[3]

C. Gerhardt, Flow of nonconvex hypersurfaces into spheres, J. Differential Geom., 32 (1990), 299-314.   Google Scholar

[4]

G. Huisken and T. Ilmanen, The Riemannian Penrose inequality, Internat. Math. Res. Notices, 20 (1997), 1045-1058.  doi: 10.1155/S1073792897000664.  Google Scholar

[5]

G. Huisken and T. Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differential Geom., 59 (2001), 353-437.   Google Scholar

[6]

G. Huisken and T. Ilmanen, Higher regularity of the inverse mean curvature flow, J. Differential Geom., 80 (2008), 433-451.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Shota Sato, Eiji Yanagida. Forward self-similar solution with a moving singularity for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 313-331. doi: 10.3934/dcds.2010.26.313
[2]

Keisuke Takasao. Existence of weak solution for mean curvature flow with transport term and forcing term. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2655-2677. doi: 10.3934/cpaa.2020116
[3]

Daehwan Kim, Juncheol Pyo. Existence and asymptotic behavior of helicoidal translating solitons of the mean curvature flow. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5897-5919. doi: 10.3934/dcds.2018256
[4]

Weronika Biedrzycka, Marta Tyran-Kamińska. Self-similar solutions of fragmentation equations revisited. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 13-27. doi: 10.3934/dcdsb.2018002
[5]

Marco Cannone, Grzegorz Karch. On self-similar solutions to the homogeneous Boltzmann equation. Kinetic & Related Models, 2013, 6 (4) : 801-808. doi: 10.3934/krm.2013.6.801
[6]

Adrien Blanchet, Philippe Laurençot. Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion. Communications on Pure & Applied Analysis, 2012, 11 (1) : 47-60. doi: 10.3934/cpaa.2012.11.47
[7]

Chiara Corsato, Franco Obersnel, Pierpaolo Omari, Sabrina Rivetti. On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space. Conference Publications, 2013, 2013 (special) : 159-169. doi: 10.3934/proc.2013.2013.159
[8]

Sigurd Angenent. Formal asymptotic expansions for symmetric ancient ovals in mean curvature flow. Networks & Heterogeneous Media, 2013, 8 (1) : 1-8. doi: 10.3934/nhm.2013.8.1
[9]

Elias M. Guio, Ricardo Sa Earp. Existence and non-existence for a mean curvature equation in hyperbolic space. Communications on Pure & Applied Analysis, 2005, 4 (3) : 549-568. doi: 10.3934/cpaa.2005.4.549
[10]

Rostislav Grigorchuk, Volodymyr Nekrashevych. Self-similar groups, operator algebras and Schur complement. Journal of Modern Dynamics, 2007, 1 (3) : 323-370. doi: 10.3934/jmd.2007.1.323
[11]

Christoph Bandt, Helena PeÑa. Polynomial approximation of self-similar measures and the spectrum of the transfer operator. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4611-4623. doi: 10.3934/dcds.2017198
[12]

Anna Chiara Lai, Paola Loreti. Self-similar control systems and applications to zygodactyl bird's foot. Networks & Heterogeneous Media, 2015, 10 (2) : 401-419. doi: 10.3934/nhm.2015.10.401
[13]

D. G. Aronson. Self-similar focusing in porous media: An explicit calculation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1685-1691. doi: 10.3934/dcdsb.2012.17.1685
[14]

G. A. Braga, Frederico Furtado, Vincenzo Isaia. Renormalization group calculation of asymptotically self-similar dynamics. Conference Publications, 2005, 2005 (Special) : 131-141. doi: 10.3934/proc.2005.2005.131
[15]

Qiaolin He. Numerical simulation and self-similar analysis of singular solutions of Prandtl equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 101-116. doi: 10.3934/dcdsb.2010.13.101
[16]

F. Berezovskaya, G. Karev. Bifurcations of self-similar solutions of the Fokker-Plank equations. Conference Publications, 2005, 2005 (Special) : 91-99. doi: 10.3934/proc.2005.2005.91
[17]

Bendong Lou. Self-similar solutions in a sector for a quasilinear parabolic equation. Networks & Heterogeneous Media, 2012, 7 (4) : 857-879. doi: 10.3934/nhm.2012.7.857
[18]

Shota Sato, Eiji Yanagida. Singular backward self-similar solutions of a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 897-906. doi: 10.3934/dcdss.2011.4.897
[19]

L. Olsen. Rates of convergence towards the boundary of a self-similar set. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 799-811. doi: 10.3934/dcds.2007.19.799
[20]

Marek Fila, Michael Winkler, Eiji Yanagida. Convergence to self-similar solutions for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 703-716. doi: 10.3934/dcds.2008.21.703

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (79)
  • HTML views (143)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences