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Sloshing, Steklov and corners: Asymptotics of sloshing eigenvalues

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Abstract

In the present paper we develop an approach to obtain sharp spectral asymptotics for Steklov type problems on planar domains with corners. Our main focus is on the two-dimensional sloshing problem, which is a mixed Steklov—Neumann boundary value problem describing small vertical oscillations of an ideal fluid in a container or in a canal with a uniform cross-section. We prove a two-term asymptotic formula for sloshing eigenvalues. In particular, this confirms a conjecture posed by Fox and Kuttler in 1983. We also obtain similar eigenvalue asymptotics for other related mixed Steklov type problems, and discuss applications to the study of Steklov spectral asymptotics on polygons.

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References

  1. M. S. Agranovich, On a mixed Poincare-Steklov type spectral problem in a Lipschitz domain, Russ. J. Math. Phys. 13 (2006), 281–290.

    Article  MathSciNet  Google Scholar 

  2. G. Alker, The scattering of short waves by a cylinder, J. Fluid Mech. 82 (1977), 673–686.

    Article  MathSciNet  Google Scholar 

  3. R. Bañuelos and T. Kulczycki, The Cauchy process and the Steklov problem, J. Funct. Anal. 211 (2004), 355–423.

    Article  MathSciNet  Google Scholar 

  4. R. Bañuelos, T. Kulczycki, I. Polterovich and B. Siudeja, Eigenvalue inequalities for mixed Steklov problems, in Operator Theory and its Applications, American Mathematical Society, Providence, RI, 2010, pp. 19–34.

    Google Scholar 

  5. R. Brown, The mixed problem for Laplace’s equation in a class of Lipschitz domains, Comm. Partial Differential Equations 19 (1994), 1217–1233.

    Article  MathSciNet  Google Scholar 

  6. L. Buhovski, Private communication, 2017.

  7. A. M. J. Davis, Two-dimensional oscillations in a a canal of arbitrary cross-section, Proc. Cambridge Phil. Soc. 61 (1965), 827–846.

    Article  MathSciNet  Google Scholar 

  8. A. M. J. Davis, Short surface waves in a canal; dependence of frequency on the curvatures and their derivatives, Quart. J. Mech. Appl. Math. 27 (1974), 523–535.

    Article  MathSciNet  Google Scholar 

  9. A. M. J. Davis and P. D. Weidman, Asymptotic estimates for two-dimensional sloshing modes, Phys. Fluids 12 (2000), 971–978.

    Article  MathSciNet  Google Scholar 

  10. U. T. Ehrenmark, Far field asymptotics of the two-dimensional linearised sloping beach problem, SIAM J. Appl. Math. 47 (1987), 965–981.

    Article  MathSciNet  Google Scholar 

  11. D. W. Fox and J. R. Kuttler, Sloshing frequencies, Z. Angew. Math. Phys. 34 (1983), 668–696.

    Article  MathSciNet  Google Scholar 

  12. D. Fischer, Proving that an orthonormal system close to a basis is also a basis, https://math.stackexchange.com/q/1365967.

  13. A. Girouard, J. Lagacé, I. Polterovich and A. Savo, The Steklov spectrum of cuboids, Mathematika, 65 (2019), 272–310.

    Article  MathSciNet  Google Scholar 

  14. A. Girouard and I. Polterovich, The Steklov eigenvalue problem: some open questions, CMS Notes 48 (2016), 16–17.

    MathSciNet  Google Scholar 

  15. A. Girouard and I. Polterovich, Spectral geometry of the Steklov problem, J. Spectr. Theory 7(2) (2017), 321–359.

    Article  MathSciNet  Google Scholar 

  16. A. G. Greenhill, Wave motion in hydrodynamics, Amer. J. Math. 9 (1887), 62–112.

    Article  MathSciNet  Google Scholar 

  17. P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, MA, 1985.

    MATH  Google Scholar 

  18. E. T. Hanson, The theory of ship waves, Proc. Roy. Soc. London Ser. A 111 (1926), 491–529.

    Article  Google Scholar 

  19. A. Hassannezhad and A. Laptev, Eigenvalue bounds of mixed Steklov problems, Commun. Contemp. Math. 22 (2020), Article no. 1950008.

  20. F. Hecht, New development in FreeFem+ +, J. Numer. Math. 20 (2012), 251–265.

    Article  MathSciNet  Google Scholar 

  21. R. A. Ibrahim, Liquid Sloshing Dynamics: Theory and Applications, Cambridge University Press, Cambridge, 2005.

    Book  Google Scholar 

  22. V. Ivrii, Spectral asymptotics for Dirichlet to Neumann operator in the domains with edges, in Microlocal Analysis, Sharp Spectral Asymptotics and Applications V, Springer, Cham, 2019, pp. 513–539.

    Chapter  Google Scholar 

  23. R. Kashaev, Private communication, 2016.

  24. R. Kashaev, V. Mangazeev and Yu. Stroganov, Star-square and tetrahedron equations in the Baxter—Bazhanov model, Int. J. Mod. Phys. A 8 (1993), 1399–1409.

    Article  MathSciNet  Google Scholar 

  25. J. Kazdan, Perturbation of complete orthonormal sets and eigenfunction expansions, Proc. Amer. Math. Soc. 27 (1971), 506–510.

    MathSciNet  MATH  Google Scholar 

  26. V. A. Kondrat’ev, Boundary value problems for elliptic equations in domains with conical or angular points, Tr. Mosk. Mat. Obs. 16 (1967), 209–292.

    MathSciNet  Google Scholar 

  27. N. D. Kopachevsky and S. G. Krein, Operator Approach to Linear Problems of Hydrodynamics, Birkhäuser, Basel, 2001.

    Book  Google Scholar 

  28. V. Kozlov, N. Kuznetsov and O. Motygin, On the two-dimensional sloshing problem, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 (2004), 2587–2603.

    Article  MathSciNet  Google Scholar 

  29. S. Krymski, M. Levitin, L. Parnovski, I. Polterovich and D. A. Sher, Inverse Steklov spectral problem for curvilinear polygons, Int. Math. Res. Not. 2021 (2021), 1–37.

    Article  MathSciNet  Google Scholar 

  30. P. Kuchment and L. Kunyansky, Differential operators on graphs and photonic crystals, Adv. Comput. Math. 16 (2002), 263–290.

    Article  MathSciNet  Google Scholar 

  31. N. Kuznetsov, T. Kulczycki, M. Kwasnicki, A. Nazarov, S. Poborchi, I. Polterovich and B. Siudeja, The legacy of Vladimir Andreevich Steklov, Notices Amer. Math. Soc. 61 (2014), 9–22.

    Article  MathSciNet  Google Scholar 

  32. N. Kuznetsov, V. Maz’ya and B. Vainberg, Linear Water Waves, Cambridge University Press, Cambridge, 2002.

    Book  Google Scholar 

  33. H. Lamb, Hydrodynamics, 1st ed., Cambridge, 1879; 6th ed., 1932; Reprinted by Dover, New York, 1945.

  34. M. Levitin, L. Parnovski, I. Polterovich and D. A. Sher, Sloshing, Steklov and corners: Asymptotics of Steklov eigenvalues for curvilinear polygons, arXiv:1908.06455 [math.SP].

  35. H. Lewy, Water waves on sloping beaches, Bull. Amer. Math. Soc. 52 (1946), 737–775.

    Article  MathSciNet  Google Scholar 

  36. M. A. Naimark, Linear Differential Operators, Dover, New York, 1967–1968.

    MATH  Google Scholar 

  37. K.-M. Perfekt and M. Putinar, Spectral bounds for the Neumann—Poincaré operator on planar domains with corners, J. Anal. Math. 124 (2014), 39–57.

    Article  MathSciNet  Google Scholar 

  38. A. S. Peters, The effect of a floating mat on water waves, Comm. Pure Appl. Math. 3 (1950), 319–354.

    Article  MathSciNet  Google Scholar 

  39. L. Rondi, Continuity properties of Neumann-to-Dirichlet maps with respect to the H-convergence of the coefficient matrices, Inverse Problems 31 (2015), Article no. 045002.

  40. L. Sandgren, A vibration problem, Medd. Lunds Univ. Math. Sem. 13 (1955), 1–84.

    MathSciNet  MATH  Google Scholar 

  41. J. J. Stoker, Surface waves in water of variable depth, Quart. Appl. Math. 5 (1947), 1–54.

    Article  MathSciNet  Google Scholar 

  42. F. Ursell, Short surface waves in a canal: dependence of frequency on curvature, J. Fluid Mech. 63 (1974), 177–181.

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Reading, Whiteknights, PO Box 220, Reading, RG6 6AX, UK

    Michael Levitin

  2. Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UK

    Leonid Parnovski

  3. Département de mathématiques et de statistique, Université de Montréal, CP 6128, succ Centre-Ville, Montréal, QC, H3C 3J7, Canada

    Iosif Polterovich

  4. Department of Mathematical Sciences, DePaul University, 2320 N Kenmore Ave, Chicago, IL, 60614, USA

    David A. Sher

Authors
  1. Michael Levitin
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  2. Leonid Parnovski
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  3. Iosif Polterovich
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  4. David A. Sher
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Correspondence to Michael Levitin.

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Levitin, M., Parnovski, L., Polterovich, I. et al. Sloshing, Steklov and corners: Asymptotics of sloshing eigenvalues. JAMA 146, 65–125 (2022). https://doi.org/10.1007/s11854-021-0188-x

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  • DOI: https://doi.org/10.1007/s11854-021-0188-x

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