关于弱中间三调和问题的谱不稳定性
On the spectral instability for weak intermediate triharmonic problems
摘要：
我们定义了三调和算子$- Δ^3$的弱中间边界条件。我们分析了这类边界条件对区域扰动的敏感性。我们构造了一个扰动$(Ω_ε)_{ε> 0}$的光滑域$Ω$的$\mathbb{R}^N$，其中$\partial Ω_ε$上的弱中间边界条件不保持在$\partial Ω$上的极限，类似于Babuška悖论的铰接板。根据$\partial Ω_ε$到$\partial Ω$的收敛性，可以在极限中产生四种不同的边界条件。在一个特定的情况下，我们得到一个“奇怪”的边界条件，具有微观能量项的形状接近域。我们的分析的许多方面可以推广到任意阶椭圆微分算子的顺序$2m$和更一般的域扰动。
英文摘要：
We define the weak intermediate boundary conditions for the triharmonic operator $- Δ^3$. We analyse the sensitivity of this type of boundary conditions upon domain perturbations. We construct a perturbation $(Ω_ε)_{ε> 0}$ of a smooth domain $Ω$ of $\mathbb{R}^N$ for which the weak intermediate boundary conditions on $\partial Ω_ε$ are not preserved in the limit on $\partial Ω$, analogously to the Babuška paradox for the hinged plate. Four different boundary conditions can be produced in the limit, depending on the convergence of $\partial Ω_ε$ to $\partial Ω$. In one particular case, we obtain a ``strange'' boundary condition featuring a microscopic energy term related to the shape of the approaching domains. Many aspects of our analysis could be generalised to an arbitrary order elliptic differential operator of order $2m$ and to more general domain perturbations.
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