论文标题
在弱中间三峰问题的光谱不稳定性上
On the spectral instability for weak intermediate triharmonic problems
论文作者
论文摘要
我们定义了三臂运算符$-Δ^3 $的弱中间边界条件。我们分析了这种类型的边界条件对域扰动的敏感性。我们构建了一个光滑域的$（ω__）_ {ε> 0} $ $ \ mathbb {r}^n $的$ω$ of $ \ partialω_ε$在$ \ partialω__$上较弱的边界条件的限制不得保留在$ \ partialω$上，以babuškapaladox the $ \ partialω$。可以在限制中产生四个不同的边界条件，这取决于$ \ partial的ω__$对$ \ partialω$的收敛性。在一种特殊情况下，我们获得了``奇怪的''边界条件，该条件具有与接近域的形状相关的微观能量术语。我们的分析的许多方面都可以推广到订单$ 200M $的任意订单椭圆机差异操作员以及更一般的域扰动。
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.
