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Article Dans Une Revue Journal de Mathématiques Pures et Appliquées Année : 2006

A nonlinear Korn inequality on a surface

Résumé

Let ω be a domain in R^2 and let θ : ω → R^3 be a smooth immersion. The main purpose of this paper is to establish a “nonlinear Korn inequality on the surface θ(ω)”, asserting that, under ad hoc assumptions, the H^1(ω)-distance between the surface θ(ω) and a deformed surface is “controlled” by the L^1(ω)-distance between their fundamental forms. Naturally, the H^1(ω)-distance between the two surfaces is only measured up to proper isometries of R^3. This inequality implies in particular the following interesting per se sequential continuity property for a sequence of surfaces. Let θ_k : ω → R^3 , k=1,2,..., be mappings with the following properties: They belong to the space H^1(ω); the vector fields normal to the surfaces θ_k(ω), k=1,2,..., are well defined a.e. in ω and they also belong to the space H^1(ω); the principal radii of curvature of the surfaces θ_k(ω), k=1,2,..., stay uniformly away from zero; and finally, the fundamental forms of the surfaces θ_k(ω) converge in L^1(ω) toward the fundamental forms of the surface θ(ω) as k → ∞. Then, up to proper isometries of R^3, the surfaces θ_k(ω) converge in H^1(ω) toward the surface θ(ω) as k → ∞. Such results have potential applications to nonlinear shell theory, the surface θ(ω) being then the middle surface of the reference configuration of a nonlinearly elastic shell.

Dates et versions

hal-00112619 , version 1 (09-11-2006)

Identifiants

Citer

Philippe Ciarlet, Liliana Gratie, Cristinel Mardare. A nonlinear Korn inequality on a surface. Journal de Mathématiques Pures et Appliquées, 2006, 9, pp.2-16. ⟨10.1016/j.matpur.2005.10.010⟩. ⟨hal-00112619⟩
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