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J. Evol. Equ. Journal of Evolution © 2019 Springer Nature Switzerland AG Equations https://doi.org/10.1007/s00028-019-00555-z L -spectral independence of Neumann Laplacians on horn-shaped domains Kouhei Matsuura Abstract. In this paper, we study spectral properties of Neumann Laplacians on horn-shaped domains. We mainly use probabilistic arguments to provide a sufficient condition for the L -spectrum being independent of p ∈[1, ∞]. 1. Introduction Let H be a positive smooth function on [1, ∞). We define a domain D ⊂ R , d ≥ 2 by 1 d−1 D ={(x , x˜ ) ∈ R × R | x > 1, |˜x | d−1 < H (x )}, d−1 where |·| d−1 is the Euclidean norm on R . D is often called a horn-shaped 2 2 domain. Let m be the Lebesgue measure on D and (L , Dom(L )) be the (non- positive) Neumann Laplacian on L (D, m). See Sect. 2 for the precise definition. The 2 2 spectral property of (L , Dom(L )) is determined by H . Evans and Harris prove in 2 2 [6](seealso[5]) that the spectrum of (L , Dom(L )) has no essential spectrum if and only if t ∞ −(d−1) d−1 lim
Journal of Evolution Equations – Springer Journals
Published: Dec 12, 2019
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