|
Brief communications
Quasi-Weyl Asymptotics of the Spectrum of the Vector Dirichlet Problem
A. S. Andreev Popov Higher Naval Academy of Radio Electronics
Abstract:
In a space of vector functions, we consider the spectral problem μAu=Pu, u=u(x), where A=(Ajk), j,k=1,…,n, Ajk=∑αaαjkD2α, P=(pjk), A⩾c0>0, P=P∗, the aαjk and pjk 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)∼(mesmΩ)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.
Keywords:
quasi-Weyl asymptotics, Dirichlet problem, vector Dirichlet problem, nonelliptic differential operator, Weyl formula, Weyl asymptotics.
Received: 13.06.2006
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
Linking options:
https://www.mathnet.ru/eng/faa2904https://doi.org/10.4213/faa2904 https://www.mathnet.ru/eng/faa/v42/i2/p75
|
Statistics & downloads: |
Abstract page: | 442 | Full-text PDF : | 199 | References: | 90 | First page: | 8 |
|