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 $\tilde{P}=P+P^{\prime }+P^{\prime \prime }$, where $P^{\prime }$ is an operator of order $<\min \{2a,a+\frac{1}{2}\}$ with certain mapping properties, and $P^{\prime \prime }$ is bounded in $L_2(\Omega )$ (e.g. $P^{\prime \prime }=V(x)\in L_{\infty }(\Omega )$). Also the regularity of eigenfunctions of $P_D$ is discussed.

Minor adjustments, 19 pages, pages, to appear in Mathematica Scandinavica