Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4615673 | Journal of Mathematical Analysis and Applications | 2015 | 19 Pages |
Abstract
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 aâR+ 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 AÏ,Σ+ in L2(Ω) of mixed problems for second-order strongly elliptic symmetric differential operators A on a bounded smooth set ΩâRn. Here the boundary âΩ=Σ is partitioned smoothly into Σ=ΣââªÎ£+, the Dirichlet condition γ0u=0 is imposed on Σâ, and a Neumann or Robin condition Ïu=0 is imposed on Σ+. It is shown that the Dirichlet-to-Neumann operator Pγ,Ï is principally of type 12 with factorization index 12, relative to Σ+. The above theory allows a detailed description of D(AÏ,Σ+) with singular elements outside of H¯32(Ω), and leads to a spectral asymptotic formula for the Krein resolvent difference AÏ,Σ+â1âAγâ1.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
Gerd Grubb,