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 Funktsional. Anal. i Prilozhen.: Year: Volume: Issue: Page: Find

 Funktsional. Anal. i Prilozhen., 2008, Volume 42, Issue 2, Pages 75–78 (Mi faa2904)

Brief communications

Quasi-Weyl Asymptotics of the Spectrum of the Vector Dirichlet Problem

A. S. Andreev

Abstract: In a space of vector functions, we consider the spectral problem $\mu Au=\mathcal{P}u$, $u=u(x)$, where $A=(A_{jk})$, $j,k=1,…,n$, $A_{jk}=\sum_\alpha a_{\alpha jk}D^{2\alpha}$, $\mathcal{P}=(p_{jk})$, $A\ge c_0>0$, $\mathcal{P}=\mathcal{P}^*$, the $a_{\alpha jk}$ and $p_{jk}$ are constants, $x\in\Omega$, and $\Omega$ is a bounded open set. The boundary conditions correspond to the Dirichlet problem. Let $N_\pm(\mu)$ be the positive and negative spectral counting functions. We establish the asymptotics $N_\pm(\mu)\sim(\operatorname{mes}_m\Omega)\varphi_\pm(\mu)$ as $\mu\to+0$. The functions $\varphi_\pm(\mu)$ are independent of $\Omega$. In the nonelliptic case, these asymptotics are in general different from the classical (Weyl) asymptotics.

Keywords: quasi-Weyl asymptotics, Dirichlet problem, vector Dirichlet problem, nonelliptic differential operator, Weyl formula, Weyl asymptotics

DOI: https://doi.org/10.4213/faa2904

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English version:
Functional Analysis and Its Applications, 2008, 42:2, 141–143

Bibliographic databases:

UDC: 517.98

Citation: A. S. Andreev, “Quasi-Weyl Asymptotics of the Spectrum of the Vector Dirichlet Problem”, Funktsional. Anal. i Prilozhen., 42:2 (2008), 75–78; Funct. Anal. Appl., 42:2 (2008), 141–143

Citation in format AMSBIB
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