Weyl asymptotics for fractional-order Dirichlet realizations in nonsmooth cases

Abstract

Let $P$ be a symmetric $2a$-order classical strongly elliptic pseudodifferential operator with even symbol $p(x,\xi )$ on $R^n$ ($0<a<1$), for example a perturbation of $(-\Delta )^a$. Let $\Omega \subset R^n$ be bounded, and let $P_D$ be the Dirichlet realization in $L_2(\Omega )$ defined under the exterior condition $u=0$ in $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 $$ \lambda _j(P_{D})=C(P,\Omega )j^{2a/n}+o(j^{2a/n})\text{ for }j\to \infty, $$ 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 $<\min\{2a, a+\frac12\}$ 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.