This paper is dedicated to the classical Hadamard formula for asymptotics of eigenvalues of the Dirichlet-Laplacian under perturbations of the boundary. We prove that the Hadamard formula still holds for $C^1$-domains with $C^1$-perturbations. We also derive an optimal estimate for the remainder term in the $C^{1,\alpha }$-case. Furthermore, if the boundary is merely Lipschitz, we show that the Hadamard formula is not valid.

中文翻译：

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Dirichlet Laplacian特征值的Hadamard渐近性

本文致力于边界扰动下Dirichlet-Laplacian特征值渐近性的经典Hadamard公式。我们证明哈达玛公式仍然适用于$C^{\mathrm{1个}}$-个域 $C^{\mathrm{1个}}$-摄动。我们还得出了剩余项中的最优估计$C^{\mathrm{1个}，\alpha }$-案件。此外，如果边界仅是Lipschitz，则表明Hadamard公式无效。