TY - JOUR

T1 - Weyl asymptotics for fractional-order Dirichlet realizations in nonsmooth cases

AU - Grubb, Gerd

N1 - Publisher Copyright: © 2023 Mathematica Scandinavica. All rights reserved.

PY - 2023

Y1 - 2023

N2 - Let P be a symmetric 2a-order classical strongly elliptic pseudodifferential operator with even symbol p(x, ξ) on Rn (0 < a < 1), for example a perturbation of (−Δ)a. Let Ω ⊂ Rn be bounded, and let PD be the Dirichlet realization in L2(Ω) defined under the exterior condition u = 0 in Rn \ Ω. When p(x, ξ) and Ω are C∞, it is known that the eigenvalues λj (ordered in a nondecreasing sequence for j → ∞) satisfy a Weyl asymptotic formula λj (PD) = C(P, Ω)j2a/n + o(j2a/n) for j → ∞, 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 P∼ = P + P′ + P′′, where P′ is an operator of order < min{2a, a + 21 } with certain mapping properties, and P′′ is bounded in L2(Ω) (e.g. P′′ = V (x) ∈ L∞(Ω)). Also the regularity of eigenfunctions of PD is discussed.

AB - Let P be a symmetric 2a-order classical strongly elliptic pseudodifferential operator with even symbol p(x, ξ) on Rn (0 < a < 1), for example a perturbation of (−Δ)a. Let Ω ⊂ Rn be bounded, and let PD be the Dirichlet realization in L2(Ω) defined under the exterior condition u = 0 in Rn \ Ω. When p(x, ξ) and Ω are C∞, it is known that the eigenvalues λj (ordered in a nondecreasing sequence for j → ∞) satisfy a Weyl asymptotic formula λj (PD) = C(P, Ω)j2a/n + o(j2a/n) for j → ∞, 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 P∼ = P + P′ + P′′, where P′ is an operator of order < min{2a, a + 21 } with certain mapping properties, and P′′ is bounded in L2(Ω) (e.g. P′′ = V (x) ∈ L∞(Ω)). Also the regularity of eigenfunctions of PD is discussed.

U2 - 10.7146/math.scand.a-138002

DO - 10.7146/math.scand.a-138002

M3 - Journal article

AN - SCOPUS:85176915777

VL - 129

SP - 593

EP - 612

JO - Mathematica Scandinavica

JF - Mathematica Scandinavica

SN - 0025-5521

IS - 3

ER -