JournalsjstVol. 3, No. 3pp. 293–316

Convergence of Dirichlet eigenvalues for elliptic systems on perturbed domains

  • Justin L. Taylor

    Murray State University, USA
Convergence of Dirichlet eigenvalues for elliptic systems on perturbed domains cover
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Abstract

We consider the eigenvalues of an elliptic operator

(Lu)β=xj(aijαβuαxi),β=1,,m,(Lu)^{\beta}=-\frac{\partial}{\partial x_j} (a^{\alpha \beta}_{ij}\frac{\partial u^{\alpha}}{\partial x_i}),\quad \beta=1,\ldots,m,

where{\verone} u=(u1,um)tu=(u^1\ldots,u^m)^t is a vector valued function and aαβ(x)a^{\alpha \beta}(x) are (n×n)(n \times n) matrices whose elements aijαβ(x)a^{\alpha \beta}_{ij}(x) are at least uniformly bounded measurable real-valued functions such that

aijαβ(x)=ajiβα(x)a^{\alpha \beta}_{ij}(x)=a^{\beta \alpha}_{ji}(x)

for any combination of α,β,i,\alpha, \beta, i, and jj. We assume we have two non-empty, open, disjoint, and bounded sets, Ω\Omega and Ω~\tilde{\Omega}, in Rn\mathbb{R}^n, and add a set TεT_{\varepsilon} of small measure to form the domain Ωε\Omega_{\varepsilon}. Then we show that as ε0+\varepsilon \rightarrow 0^+, the Dirichlet eigenvalues corresponding to the family{\verone} of domains {Ωε}ε>0\{\Omega_{\varepsilon}\}_{\varepsilon>0} converge to the Dirichlet eigenvalues corresponding to Ω0=ΩΩ~\Omega_0=\Omega \cup \tilde{\Omega}. Moreover, our rate of convergence is independent of the eigenvalues. In this paper, we consider the Lam\'{e} system, systems which satisfy a strong ellipticity condition, and systems which satisfy a Legendre--Hadamard ellipticity condition.

Cite this article

Justin L. Taylor, Convergence of Dirichlet eigenvalues for elliptic systems on perturbed domains. J. Spectr. Theory 3 (2013), no. 3, pp. 293–316

DOI 10.4171/JST/46