Weyl asymptotics for fractional-order Dirichlet realizations in nonsmooth cases

Abstract

Let $P$ be a symmetric $2a$-order classical strongly elliptic pseudodifferential operator with \emph{even} symbol $p(x,\xi)$ on $\mathbb{R}^n $ ($0<a<1$), for example a perturbation of $(-\Delta )^a$. Let $\Omega \subset \mathbb{R}^n$ be bounded, and let $P_D$ be the Dirichlet realization in $L_2(\Omega)$ defined under the exterior condition $u=0$ in $\mathbb{R}^n\setminus\Omega$. When $p(x,\xi)$ and $\Omega$ are $C^\infty $, it is known that the eigenvalues $\lambda_j$ (ordered in a nondecreasing sequence for $j\to \infty$) satisfy a Weyl asymptotic formula \begin{equation*} \lambda _j(P_{D})=C(P,\Omega )j^{2a/n}+o(j^{2a/n}) \text {for $j\to \infty $}, \end{equation*} with $C(P,\Omega)$ determined from the principal symbol of $P$. We now show that this result is valid for more general operators with a possibly nonsmooth $x$-dependence, over Lipschitz domains, and that it extends to $\tilde P=P+P'+P”$, where $P'$ is an operator of order $<\min\{2a, a+\frac 12\}$ with certain mapping properties, and $P”$ is bounded in $L_2(\Omega )$ (e.g. $P”=V(x)\in L_\infty(\Omega)$). Also the regularity of eigenfunctions of $P_D$ is discussed.