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Another proof of the existence of homothetic solitons of the inverse mean curvature flow

  • Shu-Yu Hsu ORCID logo EMAIL logo
Published/Copyright: January 30, 2024
Advances in Calculus of Variations
From the journal Advances in Calculus of Variations Volume 17 Issue 4

Abstract

We will give a new proof of the existence of non-compact homothetic solitons of the inverse mean curvature flow in ℝ n × ℝ , n ≥ 2 , of the form ( r , y ⁢ ( r ) ) or ( r ⁢ ( y ) , y ) , where r = | x | , x ∈ ℝ n , is the radially symmetric coordinate and y ∈ ℝ . More precisely for any 1 n < λ < 1 n - 1 and μ < 0 , we will give a new proof of the existence of a unique solution r ⁢ ( y ) ∈ C 2 ⁢ ( μ , ∞ ) ∩ C ⁢ ( [ μ , ∞ ) ) of the equation

r y ⁢ y ⁢ ( y ) 1 + r y ⁢ ( y ) 2 = n - 1 r ⁢ ( y ) - 1 + r y ⁢ ( y ) 2 λ ⁢ ( r ⁢ ( y ) - y ⁢ r y ⁢ ( y ) ) , r ⁢ ( y ) > 0 ,

in ( μ , ∞ ) which satisfies r ⁢ ( μ ) = 0 and r y ⁢ ( μ ) = lim y ↘ μ ⁡ r y ⁢ ( y ) = + ∞ . We prove that there exist constants y 2 > y 1 > 0 such that r y ⁢ ( y ) > 0 for any μ < y < y 1 , r y ⁢ ( y 1 ) = 0 , r y ⁢ ( y ) < 0 for any y > y 1 , r y ⁢ y ⁢ ( y ) < 0 for any μ < y < y 2 , r y ⁢ y ⁢ ( y 2 ) = 0 and r y ⁢ y ⁢ ( y ) > 0 for any y > y 2 . Moreover, lim y → + ∞ ⁡ r ⁢ ( y ) = 0 and lim y → + ∞ ⁡ y ⁢ r y ⁢ ( y ) = 0 .

Keywords: Inverse mean curvature flow; non-compact homothetic solitons; existence; asymptotic behavior
MSC 2020: 35K67; 35J75 53C42

Communicated by Frank Duzaar

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Received: 2022-11-06
Accepted: 2024-01-14
Published Online: 2024-01-30
Published in Print: 2024-10-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Another proof of the existence of homothetic solitons of the inverse mean curvature flow
  3. A Weierstrass extremal field theory for the fractional Laplacian
  4. Minimizing movements for anisotropic and inhomogeneous mean curvature flows
  5. A singular Yamabe problem on manifolds with solid cones
  6. Uniqueness for volume-constraint local energy-minimizing sets in a half-space or a ball
  7. Limit of solutions for semilinear Hamilton–Jacobi equations with degenerate viscosity
  8. Monotonicity of entire solutions to reaction-diffusion equations involving fractional p-Laplacian
  9. Hierarchy structures in finite index CMC surfaces
  10. No breathers theorem for noncompact harmonic Ricci flows with asymptotically nonnegative Ricci curvature
  11. Wolff potentials and local behavior of solutions to elliptic problems with Orlicz growth and measure data
  12. Asymptotic analysis of single-slip crystal plasticity in the limit of vanishing thickness and rigid elasticity
  13. On the regularity of optimal potentials in control problems governed by elliptic equations
  14. Sobolev embeddings and distance functions
  15. Effective quasistatic evolution models for perfectly plastic plates with periodic microstructure: The limiting regimes
  16. On the area-preserving Willmore flow of small bubbles sliding on a domain’s boundary
  17. Sobolev contractivity of gradient flow maximal functions
  18. Discrete approximation of nonlocal-gradient energies
  19. Symmetry and monotonicity of singular solutions to p-Laplacian systems involving a first order term
  20. Flat flow solution to the mean curvature flow with volume constraint
https://doi.org/10.1515/acv-2022-0092
Keywords for this article
Inverse mean curvature flow; non-compact homothetic solitons; existence; asymptotic behavior

Articles in the same Issue

  1. Frontmatter
  2. Another proof of the existence of homothetic solitons of the inverse mean curvature flow
  3. A Weierstrass extremal field theory for the fractional Laplacian
  4. Minimizing movements for anisotropic and inhomogeneous mean curvature flows
  5. A singular Yamabe problem on manifolds with solid cones
  6. Uniqueness for volume-constraint local energy-minimizing sets in a half-space or a ball
  7. Limit of solutions for semilinear Hamilton–Jacobi equations with degenerate viscosity
  8. Monotonicity of entire solutions to reaction-diffusion equations involving fractional p-Laplacian
  9. Hierarchy structures in finite index CMC surfaces
  10. No breathers theorem for noncompact harmonic Ricci flows with asymptotically nonnegative Ricci curvature
  11. Wolff potentials and local behavior of solutions to elliptic problems with Orlicz growth and measure data
  12. Asymptotic analysis of single-slip crystal plasticity in the limit of vanishing thickness and rigid elasticity
  13. On the regularity of optimal potentials in control problems governed by elliptic equations
  14. Sobolev embeddings and distance functions
  15. Effective quasistatic evolution models for perfectly plastic plates with periodic microstructure: The limiting regimes
  16. On the area-preserving Willmore flow of small bubbles sliding on a domain’s boundary
  17. Sobolev contractivity of gradient flow maximal functions
  18. Discrete approximation of nonlocal-gradient energies
  19. Symmetry and monotonicity of singular solutions to p-Laplacian systems involving a first order term
  20. Flat flow solution to the mean curvature flow with volume constraint
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