Spectral results for mixed problems and fractional elliptic operators

Abstract

In the first part of the paper we show Weyl type spectral asymptotic formulas for pseudodifferential operators $P_a$ of order $2a$, with type and factorization index $a\in R_+$, restricted to compact sets with boundary; this includes fractional powers of the Laplace operator. The domain and the regularity of eigenfunctions is described. In the second part, we apply this in a study of realizations $A_{χ,Σ_+}$ in $L_2(Ω)$ of mixed problems for a second-order strongly elliptic symmetric differential operator $A$ on a bounded smooth set $Ω\subset R^n$; here the boundary $\partialΩ=Σ$ is partioned smoothly into $Σ=Σ_-\cup Σ_+$, the Dirichlet condition $γ_0u=0$ 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 $P_{γ,χ}$ is principally of type $\frac12$ with factorization index $\frac12$, relative to $Σ_+$. The above theory allows a detailed description of $D(A_{χ,Σ_+})$ with singular elements outside of $H^{\frac32}(Ω)$, and leads to a spectral asymptotic formula for the Krein resolvent difference $A_{χ,Σ_+}^{-1}-A_γ^{-1}$.