Continuity in H1$-norms of surfaces in terms of the L1 norms of their fundamental forms
Résumé
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 R2, let θ :closure(ω) → R^3 be a smooth immersion, and let θ_k :closure(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) 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 → ∞.