Recovery of a surface with boundary and its continuity as a function of its two fundamental forms

International audienceIf a field A of class C^2 of positive-definite symmetric matrices of order two and a field B of class C^1 of symmetric matrices of order two satisfy together the Gauss and Codazzi-Mainardi equations in a connected and simply-connected open subset ω of R^2, then there exists an immersion θ ∈ C^3(ω;R^3), uniquely determined up to proper isometries in R^3, such that A and B are the first and second fundamental forms of the surface θ(ω). Let θ ̇ denote the equivalence class of θ modulo proper isometries in R^3 and let F : (A,B) → θ ̇ denote the mapping determined in this fashion. The first objective of this paper is to show that, if ω satisfies a certain “geodesic property” (in effect a mild regularity assumption on the boundary of ω) and if the fields A and B and their partial derivatives of order ≤ 2, resp. ≤ 1, have continuous extensions to ω, the extension of the field A remaining positive-definite on ω, then the immersion θ and its partial derivatives of order ≤ 3 also have continuous extensions to ω. The second objective is to show that, if ω satisfies the geodesic property and is bounded, the mapping F can be extended to a mapping that is locally Lipschitz-continuous with respect to the topologies of the Banach spaces C^2(ω) × C^1(ω) for the continuous extensions of the matrix fields (A,B), and C^3(ω) for the continuous extensions of the immersions θ