Startseite Mathematik Gibbons’ conjecture for quasilinear elliptic equations involving a gradient term
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Gibbons’ conjecture for quasilinear elliptic equations involving a gradient term

  • Phuong Le ORCID logo EMAIL logo
Veröffentlicht/Copyright: 1. Juni 2023
Forum Mathematicum
Aus der Zeitschrift Forum Mathematicum Band 35 Heft 5

Abstract

We prove Gibbons’ conjecture for the quasilinear elliptic equation

- Δ p ⁢ u + a ⁢ ( u ) ⁢ | ∇ ⁡ u | q = f ⁢ ( u )   in  ⁢ ℝ N ,

where N ≥ 2 , 2 ⁢ N + 2 N + 2 < p < 2 , q ≥ 1 and a and f are Lipschitz continuous functions which satisfy some relevant conditions. This conjecture states that every weak solution u ∈ C 1 ⁢ ( ℝ N ) of the equation with | u | ≤ 1 and lim x N → ± ∞ ⁡ u ⁢ ( x ′ , x N ) = ± 1 , uniformly in x ′ ∈ ℝ N - 1 , must depend only on x N and ∂ ⁡ u ∂ ⁡ x N > 0 in ℝ N . In particular, our result holds for a being non-decreasing on [ - 1 , - 1 + δ ] and on [ 1 - δ , 1 ] and f ⁢ ( u ) = | u | r ⁢ u ⁢ | 1 - u 2 | s ⁢ ( 1 - u 2 ) , where r , s , δ ≥ 0 . The main tool we use is an adaptation of the sliding method to the corresponding quasilinear elliptic operator.

Keywords: Gibbons’ conjecture; 1D symmetry; rigidity result
MSC 2020: 35J92; 35J62; 35B06; 35B50; 35B51

Communicated by Frank Duzaar

Funding statement: This research is funded by University of Economics and Law, VNU-HCM.

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Received: 2022-11-30
Revised: 2023-04-25
Published Online: 2023-06-01
Published in Print: 2023-09-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. On the arithmetic of polynomial semidomains
  3. Extra-special Leibniz superalgebras
  4. Restricted iso-minimum condition
  5. On the spectral large sieve inequality for symmetric-squares
  6. Simple 𝔰𝔩(V)-modules which are free over an abelian subalgebra
  7. Epsilon-strongly graded rings: Azumaya algebras and partial crossed products
  8. Non-weight modules over N = 1 Lie superalgebras of Block type
  9. Simpler foundations for the hyperbolic plane
  10. Characterizations of the mixed radial-angular central Campanato space via the commutators of Hardy type
  11. The fourth moment of Dirichlet L-functions along the critical line
  12. Diagonal restriction of Eisenstein series and Kudla–Millson theta lift
  13. Gibbons’ conjecture for quasilinear elliptic equations involving a gradient term
  14. Jantzen filtration of Weyl modules for general linear supergroups
  15. Integral formulas for a class of curvature PDE’s and applications to isoperimetric inequalities and to symmetry problems
https://doi.org/10.1515/forum-2022-0360
Schlagwörter für diesen Artikel
Gibbons’ conjecture; 1D symmetry; rigidity result

Artikel in diesem Heft

  1. Frontmatter
  2. On the arithmetic of polynomial semidomains
  3. Extra-special Leibniz superalgebras
  4. Restricted iso-minimum condition
  5. On the spectral large sieve inequality for symmetric-squares
  6. Simple 𝔰𝔩(V)-modules which are free over an abelian subalgebra
  7. Epsilon-strongly graded rings: Azumaya algebras and partial crossed products
  8. Non-weight modules over N = 1 Lie superalgebras of Block type
  9. Simpler foundations for the hyperbolic plane
  10. Characterizations of the mixed radial-angular central Campanato space via the commutators of Hardy type
  11. The fourth moment of Dirichlet L-functions along the critical line
  12. Diagonal restriction of Eisenstein series and Kudla–Millson theta lift
  13. Gibbons’ conjecture for quasilinear elliptic equations involving a gradient term
  14. Jantzen filtration of Weyl modules for general linear supergroups
  15. Integral formulas for a class of curvature PDE’s and applications to isoperimetric inequalities and to symmetry problems
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