## Spectral stability of the Neumann Laplacian

Venue: | J. Diff. Eq |

Citations: | 2 - 0 self |

### BibTeX

@ARTICLE{Burenkov_spectralstability,

author = {V. I. Burenkov and E. B. Davies},

title = {Spectral stability of the Neumann Laplacian},

journal = {J. Diff. Eq},

year = {},

pages = {485--508}

}

### OpenURL

### Abstract

We prove the equivalence of Hardy- and Sobolev-type inequalities, certain uniform bounds on the heat kernel and some spectral regularity properties of the Neumann Laplacian associated with an arbitrary region of finite measure in Euclidean space. We also prove that if one perturbs the boundary of the region within a uniform Hölder category then the eigenvalues of the Neumann Laplacian change by a small and explicitly estimated amount.

### Citations

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Citation Context ...(∂Ω) = inf {λ > 0 : M λ (∂Ω) < ∞}, M λ (∂Ω) = lim sup ε→0+ |∂εΩ| ε N−λ. Obviously M(∂Ω) ≤ N. However there exist Ω such that M λ (∂Ω) = ∞ for all λ ∈ (0, N), [9]. It can be proved that M(∂Ω) ≥ N − 1, =-=[12]-=-. If Ω satisfies the cone condition, then M(∂Ω) = N − 1, [12].3 Recall that a Whitney covering W of an open set Ω is a family of closed cubes Q each having edge length LQ = 2−k , k = 1, 2, ..., such ... |

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Citation Context ...) +∞ otherwise (1) as described in [7, Section 4.4]. It is well known that if Ω is bounded with continuous boundary ∂Ω then H has compact resolvent since the embedding W 1,2 (Ω) ⊂ L 2 (Ω) is compact, =-=[5]-=-. However, in general the spectrum of H may be quite wild, even for bounded regions in R 2 , [11]. These phenomena are not well understood, with the result that the Neumann Laplacian is far less studi... |

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Citation Context ...in Wk = {Q ∈ W : LQ = 2 −k }. If Ω has finite measure, then n(k) ≤ c12 Nk , where c1 > 0 is independent of k. Moreover, M λ (∂Ω) < ∞ if, and only if, n(k) ≤ c22 λk , where c2 > 0 is independent of k, =-=[13]-=-. Let 0 < γ ≤ 1, M, δ > 0, s ≥ 1 be an integer, and let {Vj} s j=1 be a family of bounded open cuboids and {λj} s j=1 be a family of rotations. We say that, for a bounded region Ω ⊂ Rn , its boundary ... |

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1 |
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Citation Context ... ∫ Ω d(x) −αp d N x ≤ c p 1|Ω| . Now it suffices to recall that, for regions Ω of finite measure, the inequality M(∂Ω) < N is equivalent to the existence of µ ∈ (0, 1) such that ∫ Ω d(x)−µ d N x < ∞, =-=[3]-=-. 3 Equivalence of the Sobolev-type inequalities to some spectral properties of Neumann Laplacian In this section we assume that Ω is a region in R N and suppose that H = −∆N acts in L 2 (Ω) subject t... |

1 | Edmunds D E, Rákosník J: Remarks on Poincaré inequalities - Brown - 1995 |

1 |
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Citation Context ...n of ∂Ω) is the following quantity where M(∂Ω) = inf {λ > 0 : M λ (∂Ω) < ∞}, M λ (∂Ω) = lim sup ε→0+ |∂εΩ| ε N−λ. Obviously M(∂Ω) ≤ N. However there exist Ω such that M λ (∂Ω) = ∞ for all λ ∈ (0, N), =-=[9]-=-. It can be proved that M(∂Ω) ≥ N − 1, [12]. If Ω satisfies the cone condition, then M(∂Ω) = N − 1, [12].3 Recall that a Whitney covering W of an open set Ω is a family of closed cubes Q each having ... |

1 |
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(Show Context)
Citation Context ...continuous boundary ∂Ω then H has compact resolvent since the embedding W 1,2 (Ω) ⊂ L 2 (Ω) is compact, [5]. However, in general the spectrum of H may be quite wild, even for bounded regions in R 2 , =-=[11]-=-. These phenomena are not well understood, with the result that the Neumann Laplacian is far less studied than the Dirichlet Laplacian. ∗ Supported by the INTAS grant 99-01080 and by the Russian Found... |