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 , 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 ) , k ⩾ 1 , 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.
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