Quasi-Weyl asymptotics of the spectrum of the vector Dirichlet problem

In a space of vector functions, we consider the spectral problem

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Open image in new windowa _αjk and p _jk are constants, x ∈ Ω, and Ω is a bounded open set. The boundary conditions correspond to the Dirichlet problem. Let N _±(μ) be the positive and negative spectral counting functions. We establish the asymptotics N _±(μ) ～ (mes_mΩ)φ_±(μ) as μ → +0. The functions φ_±(μ) are independent of Ω. In the nonelliptic case, these asymptotics are in general different from the classical (Weyl) asymptotics.