Higher order elliptic operators on variable domains. Stability results and boundary oscillations for intermediate problems (Q2013167)

In this work the spectral behavior of higher order elliptic operators under perturbation of the domain are studied. The classical problem \[ L u = f \quad\text{in}\quad\Omega \subset \mathbb{R}^n,\quad n \geq 2,\tag{1} \] is considered, as well as its weak formulation \(Q_\Omega(u,v) = \left\langle f,v \right\rangle_{L^2(\Omega)}\), \(\forall v \in V(\Omega)\), where \[ L u=(-1)^m \sum\limits_{|\alpha|=|\beta|=m} D^\alpha (A_{\alpha \beta} D^\beta u)+u,\quad Q_\Omega(u,v)=\sum\limits_{|\alpha|=|\beta|=m} \int\limits_\Omega A_{\alpha \beta} D^\alpha u D^\beta u + \int\limits_\Omega uv, \] \(A_{\alpha \beta} \) are bounded measurable real-valued functions in \(\mathbb{R}^n\) with \(A_{\alpha \beta}=A_{\beta \alpha}\) and \[ \sum\limits_{|\alpha|=|\beta|=m} A_{\alpha \beta}(x) \xi^\alpha \xi^\beta \geq 0 \] for \(x \in \mathbb{R}^n\) and all multi-indices of length \(m\), \(V(\Omega)\) is a linear subspace of the Sobolev space \(W^{m,2}(\Omega)\), which is a Banach space under the norm \(Q^\frac{1}{2}_\Omega(\cdot)\) and contains the closure of \(W^{m,2}_0(\Omega)\) in \(W^{m,2}(\Omega)\). The authors also consider a family of domains \(\{ \Omega_\epsilon \}_{0 < \epsilon \leq \epsilon_0}\) in \( \mathbb{R}^n\) which approach \(\Omega\) in some sense as \(\epsilon \to 0\). For this family a rather general condition is formulated (condition (C) in the work) which describes the way of the domain convergence of the family \(\{ \Omega_\epsilon \}_{0 < \epsilon \leq \epsilon_0}\) to the basic domain \(\Omega\). This condition (C) also guarantees the spectral convergence of the operator in \(\Omega_\epsilon\) to an appropriate limiting operator in \(\Omega\). Note that this condition is posed independently of the boundary conditions that are considered in the work. As first group of main results the authors prove general spectral stability results for (1) in the following three cases: \(V(\Omega_\epsilon) = W_0^{m,2}(\Omega_\epsilon)\) with Dirichlet boundary conditions; \(V(\Omega_\epsilon) = W^{m,2}(\Omega_\epsilon)\) with Neumann boundary conditions; \(V(\Omega_\epsilon) = W^{m,2}(\Omega_\epsilon) \cap W_0^{m,2}(\Omega_\epsilon)\) with so called intermediate boundary conditions. As an application, the case of the bi-harmonic operator with those intermediate boundary conditions which appear in the study of hinged plates is considered. In this case the spectral behavior is studied in the case when the boundary of the domain is subject to a periodic oscillatory perturbation. It is proved that there is a critical oscillatory behavior and the limit problem depends on whether we are above, below or just sitting on this critical value. More precisely, if the boundaries of the domains \(\Omega\) and \(\{ \Omega_\epsilon \}_{0 < \epsilon \leq \epsilon_0}\) have locally presentations in the form \(x_n=g(\bar{x})\) and \(x_n=g_\epsilon (\bar{x})= \epsilon^\lambda b(\bar{x} \epsilon^{-1})\), respectively, where \(\bar{x}=(x_1, \dots, x_{n-1}) \in W \subset \mathbb{R}^{n-1}\), \(b\) is a smooth and periodic function, then \(\gamma=\frac{3}{2}\) is a critical value. Then if \(\alpha > \frac{3}{2}\), the oscillations are not too strong and the limit problem has also the same intermediate boundary conditions. If \(\alpha < \frac{3}{2}\), the oscillations are too wild and the limit problem has a Dirichlet boundary condition in \(W\). In the case of the critical value, an extra term appears in the boundary condition for the limiting problem, which is interpreted as a ``strange curvature''