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New nonlinear estimates for surfaces in terms of their fundamental forms. (Nouvelles estimations pour des surfaces en fonction de leurs formes fondamentales.) (English. French summary) Zbl 1361.53004

Consider surfaces \(\theta\), \(\tilde{\theta}\) which are defined over the same domain \(\omega\) and immersed into \(\mathbb{R}^3\). Their normal vector fields are denoted by \(a_3\) and \(\tilde{a}_3\), respectively. The authors bound the sum \[ \| \tilde{\theta} - \theta \|_{W^{1,p}(\omega)} + \| a_3(\tilde{\theta}) - a_3(\theta) \|_{W^{1,p}(\omega)} \] and the infimum over all isometries \(r\) of \[ \| r \circ \tilde{\theta} - \theta \|_{W^{1,p}(\omega)} + \| a_3(r \circ \tilde{\theta}) - a_3(\theta) \|_{W^{1,p}(\omega)}. \] The bounds are given in terms of Sobolev norms on differences of first, second, and/or third fundamental forms of these surfaces and constants that depend on \(\omega\) and two parameters \(p > 1\), \(q \geq 1\) with \(p/2 \leq q \leq p\).
Estimates of this type are particularly useful in connection with the Koiter shell model of nonlinear elasticity theory where they allow to control the magnitude of the surface deformation in terms of the strain energy.
This article only provides definitions, results, and sketches of proofs. More details and extension to higher Sobolev norms are announced for a forthcoming paper.

MSC:

53A05 Surfaces in Euclidean and related spaces
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
74K25 Shells
74B20 Nonlinear elasticity
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