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

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
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### References:

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