Spectral asymptotics for Dirichlet elliptic operators with non-smooth coefficients
Yôichi Miyazaki · Osaka City University (Osaka City University) · 2009
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.