Article Dans Une Revue Advanced Nonlinear Studies Année : 2024

Stability and critical dimension for Kirchhoff systems in closed manifolds

Emmanuel Hebey
  • Fonction : Auteur

Résumé

Abstract The Kirchhoff equation was proposed in 1883 by Kirchhoff [ Vorlesungen über Mechanik , Leipzig, Teubner, 1883] as an extension of the classical D’Alembert’s wave equation for the vibration of elastic strings. Almost one century later, Jacques Louis Lions [“On some questions in boundary value problems of mathematical physics,” in Contemporary Developments in Continuum Mechanics and PDE’s , G. M. de la Penha, and L. A. Medeiros, Eds., Amsterdam, North-Holland, 1978] returned to the equation and proposed a general Kirchhoff equation in arbitrary dimension with external force term which was written as ∂ 2 u ∂ t 2 + a + b ∫ Ω | ∇ u | 2 d x Δ u = f ( x , u ) , $\frac{{\partial }^{2}u}{\partial {t}^{2}}+\left(a+b{\int }_{{\Omega}}\vert \nabla u{\vert }^{2}\mathrm{d}x\right){\Delta}u=f\left(x,u\right),$ where Δ = − ∑ ∂ 2 ∂ x i 2 ${\Delta}=-\sum \frac{{\partial }^{2}}{\partial {x}_{i}^{2}}$ is the Laplace-Beltrami Euclidean Laplacian. We investigate in this paper a closely related stationary version of this equation, in the case of closed manifolds, when u is vector valued and when f is a pure critical power nonlinearity. We look for the stability of the equations we consider, a question which, in modern nonlinear elliptic PDE theory, has its roots in the seminal work of Gidas and Spruck.

Dates et versions

hal-04817396 , version 1 (03-12-2024)

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Citer

Emmanuel Hebey. Stability and critical dimension for Kirchhoff systems in closed manifolds. Advanced Nonlinear Studies, 2024, 24 (1), pp.167-177. ⟨10.1515/ans-2022-0066⟩. ⟨hal-04817396⟩

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