2014/07/16 by Gerd Grubb · 1 citation
Mathematics · Physics and Astronomy · #math.AP #math-ph #math.FA #math.MP #math.SP #msc:35J57 #msc:35P20 #msc:35S15 #msc:58J40 #msc:58J50
paper · pdf · doi:10.1016/j.jmaa.2014.07.081
published as J. Math. Anal. Appl. 421 (2015), 1616-1634 · 21 pages, introduction expanded with more references, small improvements in formulations
arxiv created 2014/07/16 · arxiv updated 2014/11/04
In the first part of the paper we show Weyl type spectral asymptotic formulas for pseudodifferential operators Pa of order 2a, with type and factorization index a∈ 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 Aχ,Σ+ in L2(Ω) of mixed problems for a second-order strongly elliptic symmetric differential operator A on a bounded smooth set Ω⊂ Rn; here the boundary ∂Ω=Σ is partioned 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 \frac12 with factorization index \frac12, relative to Σ+. The above theory allows a detailed description of D(Aχ,Σ+) with singular elements outside of H\frac32(Ω), and leads to a spectral asymptotic formula for the Krein resolvent difference Aχ,Σ+-1-Aγ-1.