Weyl asymptotics for fractional-order Dirichlet realizations in nonsmooth cases - Grubb, Gerd (MATHEMATICA SCANDINAVICA, 2023)
Let $P$ be a symmetric $2a$-order classical strongly elliptic pseudodifferential operator with even symbol $p(x,ξ)$ on $\mathbb{R}^n $ ($0<a<1$), for example a perturbation of $(-Δ )^a$. Let $Ω \subset \mathbb{R}^n$ be bounded, and let $P_D$ be the Dirichlet realization in $L_2(Ω)$ defined under the exterior condition $u=0$ in $\mathbb{R}^n\setminusΩ$. When $p(x,ξ)$ and $Ω$ are $C^∞ $, it is known that the eigenvalues $λ_j$ (ordered in a nondecreasing sequence for $j→ ∞$) satisfy a Weyl asymptotic formula \begin{equation*} λ _j(P_{D})=C(P,Ω )j^{2a/n}+o(j^{2a/n}) \text {for $j→ ∞ $}, \end{equation*} with $C(P,Ω)$ 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 $L_2(Ω )$ (e.g. $P”=V(x)\in L_∞(Ω)$). Also the regularity of eigenfunctions of $P_D$ is discussed.