Spectral results for mixed problems and fractional elliptic operators

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:421

Article

Grubb, Gerd¹ 

[1] Univ Copenhagen, Dept Math Sci, DK-2100 Copenhagen, Denmark

关键词: Weyl asymptotic formula;    Dirichlet fractional Laplacian;    Eigenfunction regularity;    Zaremba problem;    Krein resolvent formula;    Dirichlet-to-Neumann operator;   

DOI  :  10.1016/j.jmaa.2014.07.081

来源: Elsevier

【 摘 要 】

One purpose of the paper is to show Weyl type spectral asymptotic formulas for pseudodifferential operators P-a of order 2a, with type and factorization index a is an element of R+ when restricted to a compact set with smooth boundary. The P-a 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 A(chi,Sigma+) in L-2(Omega) of mixed problems for second-order strongly elliptic symmetric differential operators A on a bounded smooth set Omega subset of R-n. Here the boundary partial derivative Omega = Sigma is partitioned smoothly into Sigma = Sigma(-) boolean OR Sigma(+), the Dirichlet condition gamma(0)u = 0 is imposed on Sigma(-), and a Neumann or Robin condition chi u = 0 is imposed on Sigma(+). It is shown that the Dirichlet-to-Neumann operator P-gamma,P-chi is principally of type 1/2 with factorization index 1/2, relative to Sigma(+). The above theory allows a detailed description of D(A(chi,Sigma+)) with singular elements outside of (H) over bar (3/2) (Omega), and leads to a spectral asymptotic formula for the Krein resolvent difference A(chi,Sigma+)(-1) - A(gamma)(-1). (c) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free