Osaka Journal of Mathematics

Spectral asymptotics for Dirichlet elliptic operators with non-smooth coefficients

Yoichi Miyazaki

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Abstract

We consider a $2m$-th-order elliptic operator of divergence form in a domain $\Omega$ of $\mathbb{R}^{n}$, assuming that the coefficients are Hölder continuous of exponent $r \in (0,1]$. For the self-adjoint operator associated with the Dirichlet boundary condition we improve the asymptotic formula of the spectral function $e(\tau^{2m},x,y)$ for $x=y$ to obtain the remainder estimate $O(\tau^{n-\theta}+\dist(x,\partial\Omega)^{-1}\tau^{n-1})$ with any $\theta \in (0,r)$, using the $L^{p}$ theory of elliptic operators of divergence form. We also show that the spectral function is in $C^{m-1,1-\varepsilon}$ with respect to $(x,y)$ for any small $\varepsilon > 0$. These results extend those for the whole space $\mathbb{R}^{n}$ obtained by Miyazaki [19] to the case of a domain.

Article information

Source
Osaka J. Math., Volume 46, Number 2 (2009), 441-460.

Dates
First available in Project Euclid: 19 June 2009

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1245415678

Mathematical Reviews number (MathSciNet)
MR2549595

Zentralblatt MATH identifier
1171.35443

Subjects
Primary: 35P20: Asymptotic distribution of eigenvalues and eigenfunctions

Citation

Miyazaki, Yoichi. Spectral asymptotics for Dirichlet elliptic operators with non-smooth coefficients. Osaka J. Math. 46 (2009), no. 2, 441--460. https://projecteuclid.org/euclid.ojm/1245415678


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