Let α∈]0,1[. Let Ωo be a bounded open domain of Rn of class C1,α. Let νΩo denote the outward unit normal to ∂Ωo. We assume that the Steklov problem Δu=0 in Ωo, ∂u∂νΩo=λu on ∂Ωo has a multiple eigenvalue λ˜ of multiplicity r. Then we consider an annular domain Ω(ϵ) obtained by removing from Ωo a small cavity of class C1,α and size ϵ>0, and we show that under appropriate assumptions each elementary symmetric function of r eigenvalues of the Steklov problem Δu=0 in Ω(ϵ), ∂u∂νΩ(ϵ)=λu on ∂Ω(ϵ) which converge to λ˜ as ϵ tend to zero, equals real a analytic function defined in an open neighborhood of (0,0) in R2 and computed at the point (ϵ,δ2,nϵlogϵ) for ϵ>0 small enough. Here νΩ(ϵ) denotes the outward unit normal to ∂Ω(ϵ), and δ2,2≡1 and δ2,n≡0 if n⩾3. Such a result is an extension to multiple eigenvalues of a previous result obtained for simple eigenvalues in collaboration with S. Gryshchuk.

中文翻译：

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带有小孔的区域中Steklov问题的多个特征值。功能分析方法

设α∈] 0,1 [。令Ωo为C1，α类Rn的有界开放域。令νΩo表示垂直于∂Ωo的向外单位。我们假定Steklov问题Δu= 0在Ωo中，∂u∂νΩo=λu在Ωo上具有多重本征值λ〜。然后，我们考虑通过从Ωo中去除C1，α类且尺寸ϵ> 0的小腔而获得的环形域Ω（ϵ），并证明在适当的假设下，Steklov问题Δu= 0的r个特征值的每个基本对称函数在Ω（ϵ）中，∂u∂νΩ（ϵ）=λΩ（ϵ）上的λu，随着ϵ趋于零，收敛到λ〜，等于实在R2中的（0,0）的开放邻域中定义的解析函数并在0> 0足够小的点（ϵ，δ2，nϵlogϵ）进行计算。此处νΩ（ϵ）表示垂直于∂Ω（ϵ）的向外单位，如果n⩾3，则δ2,2≡1和δ2，n≡0。