We consider the scaling of the optimal constant in Korn’s first inequality for elliptic and parabolic shells which was first given by Grabovsky and Harutyunyan with hints coming from the test functions constructed by Tovstik and Smirnov on the level of formal asymptotic expansions. Here, we employ the Bochner technique in Remannian geometry to remove the assumption that the middle surface of the shell is given by one single principal coordinate, in particularly, including closed elliptic shells.

中文翻译：

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抛物线和椭圆形壳的Korn不等式中的最佳厚度指数

我们考虑了Grabovsky和Harutyunyan首先给出的椭圆和抛物线形壳的Korn第一不等式中最佳常数的缩放，该暗示来自Tovstik和Smirnov在形式渐近展开水平上构造的测试函数。在这里，我们在Remannian几何中采用Bochner技术来消除以下假设：壳的中间表面由一个主坐标给出，尤其是包括封闭的椭圆壳。