Weyl asymptotics for fractional-order Dirichlet realizations in nonsmooth cases
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Publication:6413585
DOI10.7146/MATH.SCAND.A-138002arXiv2210.05605MaRDI QIDQ6413585
Publication date: 11 October 2022
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^nsetminusOmega $. When $p(x,xi )$ and $Omega $ are $C^infty $, it is known that the eigenvalues $lambda _j$ (ordered in a nondecreasing sequence for $j oinfty $) satisfy a Weyl asymptotic formula $$ lambda _j(P_{D})=C(P,Omega )j^{2a/n}+o(j^{2a/n}) ext{ for }j o infty, $$ 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 $ ilde 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.
Boundary value problems for PDEs with pseudodifferential operators (35S15) Higher-order elliptic equations (35J30)
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