Weyl asymptotics for fractional-order Dirichlet realizations in nonsmooth cases

TL;DR

This paper extends Weyl asymptotics for fractional-order elliptic operators to nonsmooth domains and operators with less regular coefficients, also analyzing eigenfunction regularity.

Contribution

It generalizes Weyl asymptotics to nonsmooth cases and operators with variable coefficients, including Lipschitz domains and lower-order perturbations.

Findings

Weyl asymptotics hold for nonsmooth operators and domains.

Eigenfunction regularity is analyzed in the nonsmooth setting.

Results include operators with lower-order perturbations and bounded potentials.

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.