Continuity in [formula omitted]-norms of surfaces in terms of the [formula omitted]-norms of their fundamental forms

Abstract

The main purpose of this Note is to show how a ‘nonlinear Korn's inequality on a surface’ can be established. This inequality implies in particular the following interesting per se sequential continuity property for a sequence of surfaces. Let ω be a domain in R 2 , let θ : ω ¯ → R 3 be a smooth immersion, and let θ k : ω ¯ → R 3 , k ⩾ 1 , 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 , 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 ) stay uniformly away from zero; and finally, the three fundamental forms of the surfaces θ k ( ω ) converge in L 1 ( ω ) toward the three 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 → ∞ . To cite this article: P.G. Ciarlet et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).