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.