Spectral results for mixed problems and fractional elliptic operators

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¯32(Ω), and leads to a spectral asymptotic formula for the Krein resolvent difference Aχ,Σ+−1−Aγ−1.