Spectral asymptotics for Dirichlet elliptic operators with non-smooth coefficients. (English) Zbl 1171.35443
Summary: 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 \(O(\tau^{n-\theta}+\text{dist}(x,\partial\Omega)^{-1}\tau^{n-1})\) 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 Y. Miyazaki [J. Funct. Anal. 214, No. 1, 132–154 (2004; Zbl 1066.35065)] to the case of a domain.
MSC:
| 35P20 | Asymptotic distributions of eigenvalues in context of PDEs |
Citations:
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