zbMATH — the first resource for mathematics

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 \(\gamma_0= \gamma\).

74K15 Membranes
35Q72 Other PDE from mechanics (MSC2000)
Full Text: DOI