\(L^p\)-spectral independence of Neumann Laplacians on horn-shaped domains

Journal of Evolution Equations volume 20, pages1269–1286(2020)Cite this article

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 \(L^p\)-spectrum being independent of \( p \in [1,\infty ]\).

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References

  1. 1.

    W. Arendt, Gaussian estimates and interpolation of the spectrum in\(L^{p}\), Differential Integral Equations, 7 (1994), 1153–1168.

    MathSciNet  MATH  Google Scholar 

  2. 2.

    K. Burdzy, Z.-Q. Chen, and D. E. Marshall, Traps for reflected Brownian motion, Math. Z. 252 (2006), 103–132.

    MathSciNet  Article  Google Scholar 

  3. 3.

    E. B. Davies, Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1990.

    Google Scholar 

  4. 4.

    E. B. Davies, \(L^p\)-spectral independence and\(L^1\)-analyticity, J. London Math. Soc. 52 (1995), 177–184.

    MathSciNet  Article  Google Scholar 

  5. 5.

    E. B. Davies, and B. Simon, Spectral properties of Neumann Laplacian of horns, Geom. Funct. Anal. 2 (1992), 105–117.

    MathSciNet  Article  Google Scholar 

  6. 6.

    W. D. Evans, and D. J. Harris, On the approximation numbers of Sobolev embeddings for irregular domains, Quart. J. Math. Oxford Ser. (2) 40 (1989), 13–42.

    MathSciNet  Article  Google Scholar 

  7. 7.

    M. Fukushima, On a strict decomposition of additive functionals for symmetric diffusion processes, Proc. Japan Acad. Ser. A Math. Sci. 70 (1994), 277–281.

    MathSciNet  Article  Google Scholar 

  8. 8.

    M. Fukushima, Y. Oshima, and M. Takeda, Dirichlet forms and symmetric Markov processes, Second revised and extended edition, De Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter & Co., Berlin, 2011.

    Google Scholar 

  9. 9.

    I. Kontoyiannis, and S. P. Meyn, Geometric ergodicity and the spectral gap of non-reversible Markov chains, Probab. Theory Related Fields 154 (2012), 327–339.

    MathSciNet  Article  Google Scholar 

  10. 10.

    P. C. Kunstmann, \(L_p\)-spectral properties of the Neumann Laplacian on horns, comets and stars, Math. Z. 242 (2002), 183–201.

    MathSciNet  Article  Google Scholar 

  11. 11.

    K. Mateusz, Reflecting Brownian motions on horns are not Feller?, Mathoverflow (2018). Available at https://mathoverflow.net/q/295642.

  12. 12.

    F. Maggi, Sets of finite perimeter and geometric variational problems, Cambridge Studies in Advanced Mathematics, vol. 135, Cambridge University Press, Cambridge, 2012. An introduction to geometric measure theory.

  13. 13.

    K. Matsuura, Doubly Feller property of Brownian motions with Robin boundary condition, to appear in Potential Anal. Available at https://link.springer.com/article/10.1007/s11118-018-09758-4 .

  14. 14.

    K. Matsuura, Transition densities of reflecting Brownian motions on Lipschitz domains, Available at arXiv:1901.00189 .

  15. 15.

    R. G. Pinsky, Transcience/recurrence for normally reflected Brownian motion in unbounded domains, Ann. Probab. 37 (2009), 676–686.

    MathSciNet  Article  Google Scholar 

  16. 16.

    J. Shin, and G. Trutnau, On the stochastic regularity of distorted Brownian motions, Trans. Amer. Math. Soc. 369 (2017), 7883–7915.

    MathSciNet  Article  Google Scholar 

  17. 17.

    M. Takeda, Compactness of symmetric Markov semi-groups and boundedness of eigenfuntions, to appear in Trans. Amer. Math. Soc. Available at https://www.ams.org/journals/tran/earlyview/tran7664/tran7664.pdf.

  18. 18.

    Y. Tawara, \(L^p\)-independence of spectral bounds of Schrödinger-type operators with non-local potentials, J. Math. Soc. Japan, 62 (2010), 767–788.

    MathSciNet  Article  Google Scholar 

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Acknowledgements

The author would like to thank Professor Mateusz Kwaśnicki for sharing with me his idea on the Feller property of reflecting Brownian motions on horn-shaped domains in a MathOverflow discussion at [11]. The author would like to thank also Professor Masayoshi Takeda for his helpful support. The author would also like to thank the referee for the helpful comments and suggestions.

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  1. Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan

    Kouhei Matsuura

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  1. Kouhei Matsuura
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Correspondence to Kouhei Matsuura.

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Matsuura, K. \(L^p\)-spectral independence of Neumann Laplacians on horn-shaped domains. J. Evol. Equ. 20, 1269–1286 (2020). https://doi.org/10.1007/s00028-019-00555-z

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Keywords

  • Reflecting Brownian motion
  • Dirichlet form
  • Neumann Laplacian
  • \(L^p\) spectrum
  • Spectral independence

Mathematics Subject Classification

  • 34K08
  • 34K10
  • 65C30