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A nonlinear Korn inequality on a surface. (English) Zbl 1094.53001

Let \(\omega\) be a domain in \(\mathbb{R}^2\) and let \(\theta:\overline\omega\to\mathbb{R}^3\) be a smooth immersion. The main purpose of this paper is to establish a “nonlinear Korn inequality on the surface \(\theta(\overline\omega)\)”, asserting that, under ad hoc assumptions, the \(H^1(\omega)\)-distance between the surface \(\theta (\overline\omega)\) and a deformed surface is “controlled” by the \(L^1 (\omega)\) distance between their fundamental forms. Naturally, the \(H^1(\omega)\)-distance between the two surfaces is only measured up to proper isometries of \(\mathbb{R}^3\). This inequality implies in particular the following sequential continuity property for a sequence of surfaces. Let \(\theta^k:\omega \to\mathbb{R}^3\), \(k\geq 1\), be mappings with the following properties: They belong to the space \(H^1(\omega)\); the vector fields normal to the surfaces \(\theta^k (\omega)\) are well defined a.e. in \(\omega\) and they also belong to the space \(H^1(\omega)\); the principal radii of curvature of the surfaces \(\theta^k (\omega)\), \(k\geq 1\), stay uniformly away from zero; and finally, the fundamental forms of the surfaces \(\theta^k(\omega)\) converge in \(L^1(\omega)\) toward the fundamental forms of the surface \(\theta(\overline\omega)\) as \(k\to \infty\). Then, up to proper isometries of \(\mathbb{R}^3\), the surfaces \(\theta^k(\omega)\) converge in \(H^1(\omega)\) toward the surface \(\theta(\overline\omega)\) as \(k\to\infty\).

MSC:

53A05 Surfaces in Euclidean and related spaces
74K25 Shells
46T05 Infinite-dimensional manifolds
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