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On the ellipticity of the middle surface of a shell and its application to the asymptotic analysis of ‘membrane shells’. (English) Zbl 0887.73041
Summary: We consider a surface $$S= \varphi(\omega)$$, where $$\omega\subset\mathbb{R}^2$$ is a bounded, connected, open set with a smooth boundary $$\gamma$$ and $$\varphi:\overline \omega\to\mathbb{R}^3$$ is a smooth map; let $$\gamma_{\alpha\beta}(\eta)$$ denote the components of the two-dimensional linearized strain tensor of $$S$$ and let $$\gamma_0\subset\gamma$$ with length $$\gamma_0>0$$. We assume that the norm $$\eta\mapsto\sum_{\alpha, \beta}|\gamma_{\alpha\beta}(\eta)|_{0,\omega}$$ in the space $$V_{\gamma_0}(\omega)= \{\eta\in H^1(\omega)\times H^1(\omega)\times L^2(\omega)$$; $$\eta_\alpha=0$$ on $$\gamma_0\}$$ is equivalent to the usual product norm on this space. We then establish that this assumption implies that the surface $$S$$ is uniformly elliptic and that we necessarily have $$\gamma_0= \gamma$$.

MSC:
 74K15 Membranes 35Q72 Other PDE from mechanics (MSC2000)
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