We perform a detailed analysis of the solvability of linear strain equations on several non-characteristic regions to obtain $L^2$ regularity solutions. As an application, the rigidity results on the strain tensor of the middle surface are implied by the $L^2$ regularity for non-characteristic regions. Finally, we obtain the optimal constant in the first Korn inequality scales like $h^{4/3}$ for hyperbolic shells, generalizing the assumption that the middle surface of the shell is given by a single principal system in the literature.

中文翻译：

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双曲曲面上的应变张量及其应用

我们对几个非特征区域上的线性应变方程的可解性进行了详细分析，以获得 ${\mathrm{大号}}^{\mathrm{2个}}$规律性解决方案。作为一种应用，中间表面的张量在刚度上的结果暗示了${\mathrm{大号}}^{\mathrm{2个}}$非特征区域的规律性。最后，我们在第一个Korn不等式量表中获得最佳常数，例如$H^{4/3}$ 对于双曲壳，可以将壳的中间表面由文献中的单个主系统给出的假设推广。