Weyl asymptotics for fractional-order Dirichlet realizations in nonsmooth cases
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Weyl asymptotics for fractional-order Dirichlet realizations in nonsmooth cases. / Grubb, Gerd.
In: Mathematica Scandinavica, Vol. 129, No. 3, 2023, p. 593-612.Research output: Contribution to journal › Journal article › Research › peer-review
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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 -
ID: 374451089