Spectral asymptotics for Dirichlet elliptic operators with non-smooth coefficients

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.