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Weyl formula with optimal remainder estimate of some elastic networks and applications. (Formule de Weyl avec reste optimal de quelques réseaux élastiques et applications.) (English) Zbl 1205.35304
The aim of this very important and useful paper is to study a network of vibrating elastic strings and Euler-Bernoulli beams. Here, the presented structures consist of finitely many interconnected flexible elements like strings, beams, plates representative of trusses, frames, solar panels, antennal mirrors. The spectral analysis of such models displays, in addition to its own mathematical interest, control and stabilization problems. On a finite network \(G\) associated with \(\Gamma\), as the union \(G= \bigcup^N_{j=1}\kappa_j\), where each edge \(\kappa_j\) is a Jordan curve in \(\mathbb{R}^m\) and is assumed to be parametrized by its arc length parameter \(x_j\), such that the parametrizations \(\pi_j: [O, l_j] \rightarrow \kappa_j: x_j \rightarrow \pi_j (x_j)\) are \(C^\nu ([O, l_j], \mathbb{R}^m)\) for all \(1 \leq j \leq N\), of length \(L\), made of edges \(\kappa_j\), identified to a real interval of length \(l_j\), \(j = 1,2,\dots,N\) (i.e., \(L=\sum^N_{i=1} l_i\)) the eigenvalue problem of the following form \(-d^2u_j|dx^2_j = \lambda u_j\), \(\kappa_j\), \(j = 1,2,\dots,N\), to satisfy the conditions:
(1)
\(u\) is continuous on \(G\), \(\sum_{j=s(i,h) \in N_i} O_{ih} u_{jx_j} (E_i)=0\) \(\forall i = 1,2,\dots,n\), \(u_{j_i} (E_i) = 0\) \(\forall i \in I_{\text{ext}}\), for the orientation matrix \(O=(O_{ih})_{n \times n}\) defined by \(O_{ih}=1\) if \(\kappa_{s(i,h)}\) is directed from \(E_i\) to \(E_h\), \(O_{ih}=-1\) if \(\kappa_{s(i,h)}\) is directed from \(E_h\) to \(E_i\), \(O_{ih} = 0\) else, and, \(d^4u_j/dx^4_j/dx^4_j= \lambda u_j, j= 1,\dots,N\);
(2)
\[ \begin{gathered} O_{ih} u_{jx^2_j} (E_i) = O_{i \kappa} u_{lx^2_i} (E_i) \quad\text{if }j=s(i,h),\;l=s(i, \kappa);\\ \sum_{j=s(i,h) \in N_i} O_{ih} u_{jx_j}, \quad (E_i)=0\quad \forall i= 1,2,\dots,n;\\ u_{j_i} (E_i) = 0, \quad u_{j_i, x_{j_i}^{(2)}} (E_i) = 0\quad \forall i \in I_{\text{ext}}, \end{gathered} \]
and \(u\) satisfies (1).
The operator \(\Delta_G\) on the Hilbert space \(H= \prod^N_{j=1} L^2 (O, l_j)\), endowed with the product norm, where \(D(\Delta_G)= \{u \in H\), \(u_j \in H^2 (O,l_j)\) satisfying (1)}, \(\Delta_G u= (-u_{jx_j^{(2)}})^N_{j=1}\) \(\forall u \in D(\Delta_G)\), is considered. The authors propose some asymptotic Weyl formula (with optimal remainder estimate) of an networks of strings and of Euler-Bernoulli beams. The counting function of eigenvalues \((let \lambda_0 < \lambda_1 \leq \lambda_2 \leq\cdots\leq \lambda_n \leq\cdots\) be the eigenvalues, repeated according to their multiplicity, of the self-adjoint operator \(\Delta_G\) on a \(C^2\)-network), \(N_{\Delta_G} (\lambda): = \# \sigma (\Delta_G) \cap (-\infty, \lambda]\), where in general \(\# A\) denotes the number of elements of \(A\).
Main result: There exists \(\lambda_0\gg 1\) such that: \(N_{\Delta_G} (\lambda) = \frac{L}{\pi}\sqrt{\lambda}+ \theta (1)\) uniformly on \(\lambda \in (\lambda_0, + \infty)\). A proof of the above equality is given.
As a consequence, the authors present a result concerning beam networks: There exists \(\lambda_0\gg 1\) such that: \(N_{\Delta^2_G} (\lambda): = \#\sigma (\Delta^2_G) \cap (-\infty, \lambda] = \frac{L}{\pi} \lambda + \theta (1)\) uniformly on \(\lambda \in (0,+\infty)\) and where \(D(\Delta^2_G)= \{u \in H, u_j \in H^4 (O,l_j)\) satisfying (1) and (2)}. Moreover, the authors method is valid for all elliptic operators on a graph. The following result is proposed: There exist \(M \in N^*\) and \(\eta > 0\) such that \(\mu_{n+M}- \mu_n \geq \eta M\), \(\forall n \geq 0\) where \(\mu_\kappa = \sqrt{\lambda_\kappa}\), \(\forall \kappa \geq0\). The initial and boundary value problem (application to a polynomial stabilization of a star-shaped network of Euler-Bernoulli beams)
\[ \begin{gathered} \frac{\partial^2 u_i}{\partial t^2} (x,t) + \frac{\partial^4 u_i}{\partial x^4} (x,t) = 0, \quad 0 < x < l_i,\;t >0, \\ u_i (l_i, t) = 0, \quad \frac{\partial^2 u_i}{\partial x^2} (l_i,t)=0,\;t>0,\\ u_i(0,t)= u_j (0,t), \quad \sum^N_{i=1} \frac{\partial u_i}{\partial x} (0,t)=0, \quad \frac{\partial^2 u_i}{\partial x^2} (0,t)= \frac{\partial^2 u_j}{\partial x^2} (0,t),\;t>0,\\ u_i (x, 0) = u_i^o (x) , \quad \frac{\partial u_i}{\partial t} (x,0)= u_i^1 (x),\;0 <x < l_i, \end{gathered}\tag{3} \]
for \(i,j = 1,\dots,N\), \(2\leq N\) and where \(u_i: [0, l_i] \times (0, +\infty) \rightarrow\mathbb R\) is the displacement of the beam of length \(l_i\), is considered. For the energy \(E(t)\) of \(u\) (the solution of (3) at instant \(t\), defined by \(E(t) = \sum^N_{i=1} \frac{1}{2} \int^{l_i}_0 (|\frac{\partial u_i}{\partial t} (x,t)|^2 + |\frac{\partial^2 u_i}{\partial x^2} (x,t)|^2) \,dx)\), the authors show the estimate \( E(t) \leq \frac{C_\varepsilon}{(t+1)^{\frac{1}{\gamma + \varepsilon}}} \| (u^o, u^1) \|^2_{{\mathcal D}({\mathcal G}_d)} \forall (u^o, u^1) \in{\mathcal D}({\mathcal G}_d)\) (\(N\) edges \({\mathcal G}=\{ \kappa_i, 1 \leq i \leq N\}\) where \(C_\varepsilon >0\) is a constant depending only on \(l_i\), \(i= 1,\dots,N\) and \(\varepsilon >0\).

MSC:
35Q74 PDEs in connection with mechanics of deformable solids
35P20 Asymptotic distributions of eigenvalues in context of PDEs
93D15 Stabilization of systems by feedback
93D20 Asymptotic stability in control theory
74K05 Strings
74K20 Plates
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