Abstract
The fundamental theorem of surface theory classically asserts that, if a field of positive-definite symmetric matrices (a αβ ) of order two and a field of symmetric matrices (b αβ ) of order two together satisfy the Gauss and Codazzi-Mainardi equations in a simply connected open subset ω of \({\mathbb{R}}^{2}\) , then there exists an immersion \({\bf \theta}:\omega \to {\mathbb{R}}^{3}\) such that these fields are the first and second fundamental forms of the surface \({\bf \theta}(\omega)\) , and this surface is unique up to proper isometries in \({\mathbb{R}}^3\) . The main purpose of this paper is to identify new compatibility conditions, expressed again in terms of the functions a αβ and b αβ , that likewise lead to a similar existence and uniqueness theorem. These conditions take the form of the matrix equation
where A 1 and A 2 are antisymmetric matrix fields of order three that are functions of the fields (a αβ ) and (b αβ ), the field (a αβ ) appearing in particular through the square root U of the matrix field \({\bf C} = \left(\begin{array}{lll} a_{11} & a_{12} & 0\\ a_{21} & a_{22} & 0\\ 0 & 0 & 1\end{array}\right).\) The main novelty in the proof of existence then lies in an explicit use of the rotation field R that appears in the polar factorization \({\bf \nabla}{\bf \Theta}={\bf RU}\) of the restriction to the unknown surface of the gradient of the canonical three-dimensional extension \({\bf \Theta}\) of the unknown immersion \({\bf \theta}\) . In this sense, the present approach is more “geometrical” than the classical one. As in the recent extension of the fundamental theorem of surface theory set out by S. Mardare [20–22], the unknown immersion \({\bf \theta}: \omega \to {\mathbb{R}}^3\) is found in the present approach to exist in function spaces “with little regularity”, such as \(W^{2,p}_{\rm loc}(\omega;{\mathbb{R}}^3)\), p > 2. This work also constitutes a first step towards the mathematical justification of models for nonlinearly elastic shells where rotation fields are introduced as bona fide unknowns.
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Ciarlet, P.G., Gratie, L. & Mardare, C. A New Approach to the Fundamental Theorem of Surface Theory. Arch Rational Mech Anal 188, 457–473 (2008). https://doi.org/10.1007/s00205-007-0094-0
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DOI: https://doi.org/10.1007/s00205-007-0094-0