Spectral stability estimates for the eigenfunctions of second order elliptic operators

Abstract

Key words Elliptic operators, Dirichlet boundary conditions, stability estimates for the eigenfunctions, perturbation of an open set, gap between linear operators MSC (2010) 47F05, 35J40, 35B30, 35P15 Stability of the eigenfunctions of nonnegative selfadjoint second-order linear elliptic operators subject to homogeneous Dirichlet boundary data under domain perturbation is investigated. Let Ω, Ω ⊂ R n be bounded open sets. The main result gives estimates for the variation of the eigenfunctions under perturbations Ω of Ω such that Ωε = {x ∈ Ω : dist(x, R n \Ω) > ε} ⊂ Ω ⊂ Ω ⊂ Ω in terms of powers of ε, where the parameter ε > 0 is sufficiently small. The estimates obtained here hold under some regularity assumptions on Ω, Ω . They are obtained by using the notion of a gap between linear operators, which has been recently extended by the authors to differential operators defined on different open sets.

On differentiability of eigenvalues of second order elliptic operators on non-smooth domains

“…There is a vast literature concerning this problem in the 20th century: Hadamard [22] in 1908, Courant and Hilbert [13] in the German edition of 1937, Polya and Szëgo [43] in 1951, Garabedian and Schiffer [19,20] in 1952-1953, Polya and Schiffer [42] in 1953, Schiffer [46] in 1954 and thereafter [2,18,3,27,39,[36][37][38]40,47,16,44,14,17,21]. For more recent advances, we cite the works [29,11,15,35,6,26,25,30,31,4,8,9,33,10,34,32]. We also mention three interesting works on generic properties of eigenvalues and eigenfunctions due to Uhlenbeck [48,49] and Pereira [41] which are closely related to the issue of this paper.…”

Sharp bounds for domain perturbations of Dirichlet Laplacians defined on smooth domains

“…Our main results are: Remark (i) Results similar to Theorems 1 and 2 have recently been proved for N = 2 and N = 3 in [2]. For example in Theorem 3.6 of [2], using very different methods, Burenkov (i = k p + 1, .…”

“…So Theorems 1 and 2 are extensions of the results in [2] to R N , N ≥ 4, with an improved estimate when the elliptic operator is the Dirichlet Laplacian. We also eliminate the smoothness assumption on ( ) in [2]. The methods in this paper are a combination of those in [8][9][10].…”

“…1. For all sufficient small > 0 we let K ( ) be an (2 ) and that K ( ) separates ∂ ( ) and ∂ (2 ) in the sense that if x ∈ ∂ ( ) and y ∈ ∂ (2 ) and is a path joining x to y, then…”

Conformal spectral stability estimates for the Neumann Laplacian

“…The spectral stability estimates for the Laplace operator were intensively studied in the last two decades. See, for example, the survey papers [12], [7] where one can found the history of the problem, main results in this area and appropriate references.…”