Weyl asymptotics for fractional-order Dirichlet realizations in nonsmooth cases


  • Gerd Grubb




Let $P$ be a symmetric $2a$-order classical strongly elliptic pseudodifferential operator with \emph{even} symbol $p(x,\xi)$ on $\mathbb{R}^n $ ($0<a<1$), for example a perturbation of $(-\Delta )^a$. Let $\Omega \subset \mathbb{R}^n$ be bounded, and let $P_D$ be the Dirichlet realization in $L_2(\Omega)$ defined under the exterior condition $u=0$ in $\mathbb{R}^n\setminus\Omega$. 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 \begin{equation*} \lambda _j(P_{D})=C(P,\Omega )j^{2a/n}+o(j^{2a/n}) \text {for $j\to \infty $}, \end{equation*} 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 $<\min\{2a, a+\frac 12\}$ 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.


Abels, H., and Grubb, G., Fractional-order operators on nonsmooth domains, J. Lond. Math. Soc. (2) 107 (2023), no. 4, 1297–1350.

Blumenthal, R. M., and Getoor, R. K., The asymptotic distribution of the eigenvalues for a class of Markov operators, Pacific J. Math. 9 (1959), 399–408. http://projecteuclid.org/euclid.pjm/1103039263

Chen, Z.-Q., and Song, R., Two-sided eigenvalue estimates for subordinate processes in domains, J. Funct. Anal. 226 (2005), no. 1, 90–113. https://doi.org/10.1016/j.jfa.2005.05.004

Daners, D., Domain perturbation for linear and semi-linear boundary value problems, Handbook of differential equations, stationary partial differential equations, vol. VI, 1–81, Elsevier/North-Holland, Amsterdam, 2008. https://doi.org/10.1016/S1874-5733(08)80018-6

Dyda, B., Kuznetsov, A., and Kwaśnicki, M., Eigenvalues of the fractional Laplace operator in the unit ball, J. Lond. Math. Soc. (2) 95 (2017), no. 2, 500–518. https://doi.org/10.1112/jlms.12024

Frank, R., Eigenvalue bounds for the fractional Laplacian: A review, Recent developments in nonlocal theory, 210–235, De Gruyter, Berlin, 2018. https://doi.org/10.1515/9783110571561-007

Geisinger, L., A short proof of Weyl's law for fractional differential operators, J. Math. Phys. 55 (2014), no. 1, 011504, 7pp. https://doi.org/10.1063/1.4861935

Gohberg, I. C., and Krein, M. G., Introduction to the theory of linear nonselfadjoint operators, Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I. 1969.

Grubb, G., Singular Green operators and their spectral asymptotics, Duke Math. J. 51 (1984), no. 3, 477–528. https://doi.org/10.1215/S0012-7094-84-05125-1

Grubb, G., Functional calculus of pseudodifferential boundary problems, Pro­gress in Math. vol. 65, Second Edition, Birkhäuser, Boston 1996. https://doi.org/10.1007/978-1-4612-0769-6

Grubb, G., Distributions and operators, Graduate Texts in Mathematics, vol. 252, Springer, New York, 2009.

Grubb, G., Spectral asymptotics for nonsmooth singular Green operators, Comm. Partial Differential Equations 39 (2014), no. 3, 530–573. https://doi.org/10.1080/03605302.2013.864207

Grubb, G., Fractional Laplacians on domains, a development of Hörmander's theory of µ-transmission pseudodifferential operators, Adv. Math. 268 (2015), 478–528. https://doi.org/10.1016/j.aim.2014.09.018

Grubb, G., Spectral results for mixed problems and fractional elliptic operators, J. Math. Anal. Appl. 421 (2015), no. 2, 1616–1634. https://doi.org/10.1016/j.jmaa.2014.07.081

Grubb, G., Limited regularity of solutions to fractional heat and Schrödinger equations, Discrete Contin. Dyn. Syst. 39 (2019), no. 6, 3609–3634. https://doi.org/10.3934/dcds.2019148

Grubb, G., Fourier methods for fractional-order operators, arXiv:2208.017175, prepared for the Proceedings of the RIMS Symposium “Harmonic Analysis and Nonlinear Partial Differential equations”, July 11–13, 2022, in the RIMS Kokyuroku Bessatsu series.

Grubb, G., Resolvents for fractional-order operators with nonhomogeneous local boundary conditions, J. Funct. Anal. 284 (2023), no. 7, Paper No. 109815, 55 pp. https://doi.org/10.1016/j.jfa.2022.109815

Hirsch, M. W., Differential Topology, Graduate Texts in Mathematics, No. 33 Springer-Verlag, New York-Heidelberg, 1976.

Ivrii, V., Spectral asymptotics for fractional Laplacians, Differential equations, mathematical physics, and applications: Selim Grigorievich Krein centennial, 159–170, Contemp. Math., 734, Amer. Math. Soc., RI, 2019. https://doi.org/10.1090/conm/734/14770

Lions, J.-L., and Magenes, E., Problèmes aux limites non homogènes et applications. Vol. 1 et 2, Éditions Dunod, Paris 1968.

Reed, M., and Simon, B., Methods of modern mathematical physics. IV. Analysis of operators, Academic Press. New York-London 1978.

Seeley, R. T., Complex powers of an elliptic operator, 1967 Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), pp. 288–307 Amer. Math. Soc., Providence, R.I.



How to Cite

Grubb, G. (2023). Weyl asymptotics for fractional-order Dirichlet realizations in nonsmooth cases. MATHEMATICA SCANDINAVICA, 129(3). https://doi.org/10.7146/math.scand.a-138002