Distribution of resonances for the Neumann problem in linear elasticity outside a strictly convex body. (English) Zbl 0846.35139

Stefanov, P.; Vodev, G.

Duke Math. J. 78, No. 3, 677-714 (1995).

Let \(\Delta_e\) be the elasticity operator defined by \[ \Delta_e v= \mu_0 \Delta v+ (\lambda_0+ \mu_0) \nabla (\nabla v), \] \(v= (v_1, v_2, v_3)\). \(\lambda_0\), \(\mu_0\) are the Lamé constants with \(\mu_0> 0\), \(3\lambda_0+ 2\mu_0> 0\). The authors consider the elasticity operator under Neumann boundary conditions of the form \[ \sum^3_{j= 1} \sigma_{ij}(v) \nu_j|_\Gamma= 0,\quad i= 1, 2, 3, \] where \(\sigma_{ij}\) is the stress tensor and \(\nu\) is the outer normal to \(\Gamma= \partial \Omega\), \(\Omega= \mathbb{R}^3\backslash {\mathcal O}\) and \(\mathcal O\) is a strictly convex compact in \(\mathbb{R}^3\) with \(C^\infty\)-smooth boundary \(\Gamma\). The operator \(-\Delta_e\) can be extended to a selfadjoint operator \(L\) on \(L^2(\Omega; C^3)\). If the cut-off resolvent \(R_\chi(\lambda)= \chi(L- \lambda^2)^{- 1} \chi\), \(\chi\in \mathbb{C}^\infty_0\) is extended to the whole complex plane then the poles are called resonances. The main result is the following:
(a) For any \(C_1> 0\), there exists a \(C_2> 0\), such that for any \(N> 0\) there are no resonances in the domain \(C_N|\lambda|^{- N}< \text{Im } \lambda< C_1\ln|\lambda|,\;|\text{Re } \lambda|> C_2\), with some \(C_N> 0\).
(b) There exist 2 infinite sequences \(\{\lambda_j\}\), \(\{- \overline\lambda_j\}\) of distinct resonances of \(L\), such that for any \(N> 0\), \(0< \text{Im } \lambda_j\leq C_N|\lambda_j|^{- N}\).

Reviewer: B.Dittmar (Halle)

MSC:

35Q72	Other PDE from mechanics (MSC2000)
74B05	Classical linear elasticity
35J15	Second-order elliptic equations

Keywords:

resonances; exterior problem; elasticity operator; Neumann boundary conditions

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