Asymptotic Analysis

We consider the Weyl formula for the asymptotic number of eigenvalues $N(\lambda)$ for some elliptic operators. We prove remainder estimates of the form $N(\lambda){\rm O}(\lambda^{-\mu})$ with $\mu$ depending on the regularity of coefficients and ranging between 0 and the optimal value in the standard situation of smooth coefficients, i.e., $0<\mu<2/m$ in the case of globally elliptic operators on ${\NBbbR}^d$ , and $0<\mu<1/m$ in the case of a smooth compact manifold without boundary, where $m$ is the degree of the considered operators.

Starting from the three‐dimensional dynamic equations of elastic shells, by the asymptotic analysis, under certain conditions we give the justification of dynamic equations of membrane shells.