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The triharmonic equation on the Heisenberg group

  • Majid Izadjoo and Mojgan Akbari EMAIL logo
Published/Copyright: February 23, 2023
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Georgian Mathematical Journal
From the journal Georgian Mathematical Journal Volume 30 Issue 3

Abstract

Consider the equation

{ - Δ H 3 ⁢ u = f ⁢ ( ξ , u , ∇ H ⁡ u , ∇ H 2 ⁡ u , ∇ H 3 ⁡ u , ∇ H 4 ⁡ u , ∇ H 5 ⁡ u ) in  ⁢ Ω , u > 0 in  ⁢ Ω , u = ∂ ∂ ⁡ ν ⁢ ( ∇ H 2 ⁡ u ) = ∂ ∂ ⁡ ν ⁢ ( ∇ H 3 ⁡ u ) = 0 on  ⁢ ∂ ⁡ Ω ,

where Ω is a domain of the finite-dimensional space ℍ n and f is a positive and bounded function. We prove the existence of a solution for the above equation. In addition, we prove the uniqueness and the cylindrical symmetry of the solution.

Keywords: Triharmonic equation; Arzelà–Ascoli theorem; Heisenberg group; sub-elliptic equation
MSC 2010: 31B30; 35H20; 35R03

References

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Received: 2022-01-31
Revised: 2022-12-04
Accepted: 2022-12-14
Published Online: 2023-02-23
Published in Print: 2023-06-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. The second cohomology spaces of 𝒦(1) with coefficients in the superspace of weighted densities and deformations of the superspace of symbols on S 1|1
  3. On completely non-Baire unions in category bases
  4. Square-free values of n 2 + n + 1
  5. The stability and convergence analysis for singularly perturbed Sobolev problems with Robin type boundary condition
  6. Sign-changing solutions for parametric Neumann problems with broken symmetry and arbitrary growth
  7. The triharmonic equation on the Heisenberg group
  8. On Busemann–Feller extensions of translation invariant density differentiation bases
  9. Multiplicative bi-skew Jordan triple derivations on prime ∗-algebra
  10. Nonmeasurable products of absolutely negligible sets in uncountable solvable groups
  11. The double Fourier transform of non-Lebesgue integrable functions of bounded Hardy–Krause variation
  12. On the stochastic integral representation of Brownian functionals
  13. Generalized Bessel matrix functions
  14. Degenerate time-fractional diffusion equation with initial and initial-boundary conditions
  15. Some new characterizations of EP elements, partial isometries and strongly EP elements in rings with involution
  16. 𝐿𝑝(⋅) − 𝐿𝑞(⋅) estimates for convolution operators with singular measures supported on surfaces of half the ambient dimension
  17. Tilting pairs and Wakamatsu tilting subcategories over triangular matrix algebras
https://doi.org/10.1515/gmj-2023-2012
Keywords for this article
Triharmonic equation; Arzelà–Ascoli theorem; Heisenberg group; sub-elliptic equation

Articles in the same Issue

  1. Frontmatter
  2. The second cohomology spaces of 𝒦(1) with coefficients in the superspace of weighted densities and deformations of the superspace of symbols on S 1|1
  3. On completely non-Baire unions in category bases
  4. Square-free values of n 2 + n + 1
  5. The stability and convergence analysis for singularly perturbed Sobolev problems with Robin type boundary condition
  6. Sign-changing solutions for parametric Neumann problems with broken symmetry and arbitrary growth
  7. The triharmonic equation on the Heisenberg group
  8. On Busemann–Feller extensions of translation invariant density differentiation bases
  9. Multiplicative bi-skew Jordan triple derivations on prime ∗-algebra
  10. Nonmeasurable products of absolutely negligible sets in uncountable solvable groups
  11. The double Fourier transform of non-Lebesgue integrable functions of bounded Hardy–Krause variation
  12. On the stochastic integral representation of Brownian functionals
  13. Generalized Bessel matrix functions
  14. Degenerate time-fractional diffusion equation with initial and initial-boundary conditions
  15. Some new characterizations of EP elements, partial isometries and strongly EP elements in rings with involution
  16. 𝐿𝑝(⋅) − 𝐿𝑞(⋅) estimates for convolution operators with singular measures supported on surfaces of half the ambient dimension
  17. Tilting pairs and Wakamatsu tilting subcategories over triangular matrix algebras
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