Spectral results for mixed problems and fractional elliptic operators,

TY - JOUR

T1 - Spectral results for mixed problems and fractional elliptic operators,

AU - Grubb, Gerd

PY - 2015

Y1 - 2015

N2 - One purpose of the paper is to show Weyl type spectral asymptotic formulas for pseudodifferential operators Pa of order 2a, with type and factorization index a∈R+ when restricted to a compact set with smooth boundary. The Pa include fractional powers of the Laplace operator and of variable-coefficient strongly elliptic differential operators. Also the regularity of eigenfunctions is described. The other purpose is to improve the knowledge of realizations Aχ,σ+ in L2(Ω) of mixed problems for second-order strongly elliptic symmetric differential operators A on a bounded smooth set Ω⊂Rn. Here the boundary ∂Ω=σ is partitioned smoothly into σ=σ-∪σ+, the Dirichlet condition γ0u=0 is imposed on σ-, and a Neumann or Robin condition χu=0 is imposed on σ+. It is shown that the Dirichlet-to-Neumann operator Pγ,χ is principally of type 12 with factorization index 12, relative to σ+. The above theory allows a detailed description of D(Aχ,σ+) with singular elements outside of H 3/2(Ω), and leads to a spectral asymptotic formula for the Krein resolvent difference Aχ,σ+-1-Aγ-1.

AB - One purpose of the paper is to show Weyl type spectral asymptotic formulas for pseudodifferential operators Pa of order 2a, with type and factorization index a∈R+ when restricted to a compact set with smooth boundary. The Pa include fractional powers of the Laplace operator and of variable-coefficient strongly elliptic differential operators. Also the regularity of eigenfunctions is described. The other purpose is to improve the knowledge of realizations Aχ,σ+ in L2(Ω) of mixed problems for second-order strongly elliptic symmetric differential operators A on a bounded smooth set Ω⊂Rn. Here the boundary ∂Ω=σ is partitioned smoothly into σ=σ-∪σ+, the Dirichlet condition γ0u=0 is imposed on σ-, and a Neumann or Robin condition χu=0 is imposed on σ+. It is shown that the Dirichlet-to-Neumann operator Pγ,χ is principally of type 12 with factorization index 12, relative to σ+. The above theory allows a detailed description of D(Aχ,σ+) with singular elements outside of H 3/2(Ω), and leads to a spectral asymptotic formula for the Krein resolvent difference Aχ,σ+-1-Aγ-1.

KW - Faculty of Science

KW - matematik

U2 - 10.1016/j.jmaa.2014.07.081

DO - 10.1016/j.jmaa.2014.07.081

M3 - Journal article

SN - 0022-247X

VL - 421

SP - 1616

EP - 1634

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 2

ER -