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The spectrum asymptotics for the Dirichlet problem in the case of the biharmonic operator in a domain with highly indented boundary. (English. Russian original) Zbl 1232.31001

St. Petersbg. Math. J. 22, No. 6, 941-983 (2011); translation from Algebra Anal. 22, No. 6, 127-184 (2010).
The paper deals with the spectral problem
\[ \begin{cases}\Delta_x^2u^{\varepsilon}(x)=\lambda^{\varepsilon}u^{\varepsilon}(x),\,\,\,x\in\Omega^{\varepsilon},\\ u^{\varepsilon}(x)=0,\,\,\,\partial_{n^{\varepsilon}}u^{\varepsilon}(x)=0,\,\,\,x\in \Gamma^{\varepsilon},\end{cases}(P) \]
where the perturbed domain \(\Omega^{\varepsilon}\subset\mathbb{R}^2\) has a rapidly oscillating boundary \(\Gamma^{\varepsilon}\) defined by
\[ \Gamma^{\varepsilon}=\big\{x\in {\mathcal V}:\,\,s\in\Gamma,\,\,n=\varepsilon^{\gamma}H(\varepsilon^{-1}s,s)\big\}, \]
where \(\varepsilon=1/N\) is a small parameter, \(N\) is a large positive integer, \(\gamma\) is a quantity measuring the “irregularity” of the boundary, and \(H\) is a profile function that is smooth relative to both variables \(s\) and \(\eta=\varepsilon^{-1}s\) and \(1\)-periodic relative to \(\eta\). Here \(\Delta_x\) is the Laplace operator in the Cartesian coordinates \(x=(x_1,x_2)\), \(\lambda^{\varepsilon}\) is the spectral parameter, and \(\partial_{n^{\varepsilon}}\) denotes the derivative along the outward normal to the boundary \(\Gamma^{\varepsilon}\). By using the methods of composite and matching asymptotic expansions, the authors present a detailed asymptotic analysis of the spectral problem \((P)\). The resulting formulas relate the eigenvalues in domains with close irregular boundaries and make it possible to control the order of perturbation and to find conditions ensuring the validity (or violation) of the classical Hadamard formula.

MSC:

31A25 Boundary value and inverse problems for harmonic functions in two dimensions
31A30 Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions
35P20 Asymptotic distributions of eigenvalues in context of PDEs
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