January  2020, 19(1): 113-122. doi: 10.3934/cpaa.2020007

Estimates for sums of eigenvalues of the free plate via the fourier transform

1. 

Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Università degli Studi di Napoli Federico II, Napoli, Italy

2. 

Department of Mathematics, Bucknell University, Lewisburg, Pennsylvania 17837

* Corresponding author

Received  October 2018 Revised  April 2019 Published  July 2019

We obtain estimates for sums of eigenvalues of the free plate under tension in terms of the dimension of the ambient space, the volume of the domain, and the tension parameter. We consequently obtain similar estimates for the eigenvalues. Our results generalize those of Kröger for the free membrane contained in [16].

Citation: Barbara Brandolini, Francesco Chiacchio, Jeffrey J. Langford. Estimates for sums of eigenvalues of the free plate via the fourier transform. Communications on Pure and Applied Analysis, 2020, 19 (1) : 113-122. doi: 10.3934/cpaa.2020007
References:
[1]

M. S. Ashbaugh and R. D. Benguria, On Rayleigh's conjecture for the clamped plate and its generalization to three dimensions, Duke Math. J., 78 (1995), 1-17.  doi: 10.1215/S0012-7094-95-07801-6.

[2]

M. S. Ashbaugh and R. S. Laugesen, Fundamental tones and buckling loads of clamped plates, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 23 (1996), 383-402. 

[3]

B. BrandoliniF. ChiacchioE. Dryden and J. J. Langford, Sharp Poincaré inequalities in a class of non-convex sets, J. Spectr. Theory, 8 (2018), 1583-1615.  doi: 10.4171/JST/236.

[4]

B. BrandoliniF. Chiacchio and C. Trombetti, Sharp estimates for eigenfunctions of a Neumann problem, Comm. Partial Differential Equations, 34 (2009), 1317-1337.  doi: 10.1080/03605300903089859.

[5]

B. BrandoliniF. Chiacchio and C. Trombetti, Optimal lower bounds for eigenvalues of linear and nonlinear Neumann problems, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 31-45.  doi: 10.1017/S0308210513000371.

[6]

L. M. Chasman, An isoperimetric inequality for fundamental tones of free plates, Comm. Math. Phys., 303 (2011), 421-449.  doi: 10.1007/s00220-010-1171-z.

[7]

L. M. Chasman, An isoperimetric inequality for fundamental tones of free plates with nonzero Poisson's ratio, Appl. Anal., 95 (2016), 1700-1735.  doi: 10.1080/00036811.2015.1068299.

[8]

Q. M. Cheng and G. Wei, A lower bound for eigenvalues of a clamped plate problem, Calc. Var. Partial Differential Equations, 42 (2011), 579–590. doi: 10.1007/s00526-011-0399-6.

[9]

Q. M. Cheng and G. Wei, Upper and lower bounds for eigenvalues of the clamped plate problem, J. Differential Equations, 255 (2013), 220–233. doi: 10.1016/j.jde.2013.04.004.

[10]

B. Dittmar, Sums of reciprocal eigenvalues of the Laplacian, Math. Nachr., 237 (2002), 45–61. doi: 10.1002/1522-2616(200211)245:1<45::AID-MANA45>3.0.CO;2-L.

[11]

B. Dittmar, Sums of free membrane eigenvalues, J. Anal. Math., 95 (2005), 323–332. doi: 10.1007/BF02791506.

[12]

P. Freitas and J. B. Kennedy, Summation formula inequalities for eigenvalues of Schrödinger operators, J. Spectr. Theory, 6 (2016), 483–503. doi: 10.4171/JST/130.

[13]

F. Gazzola, H.-C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems. Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains, Lecture Notes in Mathematics, 1991. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-12245-3.

[14]

E. M. Harrell and J. Stubbe, On trace identities and universal eigenvalue estimates for some partial differential operators, Trans. Amer. Math. Soc., 349 (1997), 1797-1809.  doi: 10.1090/S0002-9947-97-01846-1.

[15]

B. Kawohl, H. A. Levine and W. Velte, Buckling eigenvalues for a clamped plate embedded in an elastic medium and related questions, SIAM J. Math. Anal., 24 (1993), 327–340. doi: 10.1137/0524022.

[16]

P. Kröger, Upper bounds for the Neumann eigenvalues on a bounded domain in Euclidean space, J. Funct. Anal., 106 (1992), 353–357. doi: 10.1016/0022-1236(92)90052-K.

[17]

P. Kröger, Estimates for sums of eigenvalues of the Laplacian, J. Funct. Anal., 126 (1994), 217–227. doi: 10.1006/jfan.1994.1146.

[18]

A. Laptev, Dirichlet and Neumann eigenvalue problems on domains in Euclidean spaces, J. Funct. Anal., 151 (1997), 531–545. doi: 10.1006/jfan.1997.3155.

[19]

R. S. Laugesen and B. A. Siudeja, Sums of Laplace eigenvalues–rotationally symmetric maximizers in the plane, J. Funct. Anal., 260 (2011), 1795–1823. doi: 10.1016/j.jfa.2010.12.018.

[20]

L. Li and L. Tang, Some upper bounds for sums of eigenvalues of the Neumann Laplacian, Proc. Amer. Math. Soc., 134 (2006), 3301–3307. doi: 10.1090/S0002-9939-06-08355-9.

[21]

P. Li and S. T. Yau, On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys., 88 (1983), 309–318.

[22]

A. D. Melas, A lower bound for sums of eigenvalues of the Laplacian, Proc. Amer. Math. Soc., 131 (2003), 631–636. doi: 10.1090/S0002-9939-02-06834-X.

[23]

N. Nadirashvili, Rayleigh's conjecture on the principal frequency of the clamped plate, Arch. Ration. Mech. Anal., 129 (1995), 1-10.  doi: 10.1007/BF00375124.

[24]

Y. Safarov and D. Vassiliev, The Asymptotic Distribution of Eigenvalues of Partial Differential Operators (English summary), Translated from the Russian manuscript by the authors. Translations of Mathematical Monographs, 155. American Mathematical Society, Providence, RI, 1997.

[25]

B. A. Siudeja, Generalized tight $p$-frames and spectral bounds for Laplacian-like operators, Appl. Comput. Harmon. Anal., 42 (2017), 167-198.  doi: 10.1016/j.acha.2015.08.001.

[26]

G. Szegő, Inequalities for certain eigenvalues of a membrane of given area, J. Rational Mech. Anal., 3 (1954), 343-356.  doi: 10.1512/iumj.1954.3.53017.

[27]

G. Talenti, On the first eigenvalue of the clamped plate, Ann. Mat. Pura Appl., 129 (1981), 265-280.  doi: 10.1007/BF01762146.

[28]

S. Yıldırım Yolcu and T. Yolcu, Estimates on the eigenvalues of the clamped plate problem on domains in Euclidean spaces, J. Math. Phys., 54 (2013), 043515, 13 pp. doi: 10.1063/1.4801446.

[29]

H. Weinberger, An isoperimetric inequality for the $N$-dimensional free membrane problem, J. Rational Mech. Anal., 5 (1956), 633-636.  doi: 10.1512/iumj.1956.5.55021.

show all references

References:
[1]

M. S. Ashbaugh and R. D. Benguria, On Rayleigh's conjecture for the clamped plate and its generalization to three dimensions, Duke Math. J., 78 (1995), 1-17.  doi: 10.1215/S0012-7094-95-07801-6.

[2]

M. S. Ashbaugh and R. S. Laugesen, Fundamental tones and buckling loads of clamped plates, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 23 (1996), 383-402. 

[3]

B. BrandoliniF. ChiacchioE. Dryden and J. J. Langford, Sharp Poincaré inequalities in a class of non-convex sets, J. Spectr. Theory, 8 (2018), 1583-1615.  doi: 10.4171/JST/236.

[4]

B. BrandoliniF. Chiacchio and C. Trombetti, Sharp estimates for eigenfunctions of a Neumann problem, Comm. Partial Differential Equations, 34 (2009), 1317-1337.  doi: 10.1080/03605300903089859.

[5]

B. BrandoliniF. Chiacchio and C. Trombetti, Optimal lower bounds for eigenvalues of linear and nonlinear Neumann problems, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 31-45.  doi: 10.1017/S0308210513000371.

[6]

L. M. Chasman, An isoperimetric inequality for fundamental tones of free plates, Comm. Math. Phys., 303 (2011), 421-449.  doi: 10.1007/s00220-010-1171-z.

[7]

L. M. Chasman, An isoperimetric inequality for fundamental tones of free plates with nonzero Poisson's ratio, Appl. Anal., 95 (2016), 1700-1735.  doi: 10.1080/00036811.2015.1068299.

[8]

Q. M. Cheng and G. Wei, A lower bound for eigenvalues of a clamped plate problem, Calc. Var. Partial Differential Equations, 42 (2011), 579–590. doi: 10.1007/s00526-011-0399-6.

[9]

Q. M. Cheng and G. Wei, Upper and lower bounds for eigenvalues of the clamped plate problem, J. Differential Equations, 255 (2013), 220–233. doi: 10.1016/j.jde.2013.04.004.

[10]

B. Dittmar, Sums of reciprocal eigenvalues of the Laplacian, Math. Nachr., 237 (2002), 45–61. doi: 10.1002/1522-2616(200211)245:1<45::AID-MANA45>3.0.CO;2-L.

[11]

B. Dittmar, Sums of free membrane eigenvalues, J. Anal. Math., 95 (2005), 323–332. doi: 10.1007/BF02791506.

[12]

P. Freitas and J. B. Kennedy, Summation formula inequalities for eigenvalues of Schrödinger operators, J. Spectr. Theory, 6 (2016), 483–503. doi: 10.4171/JST/130.

[13]

F. Gazzola, H.-C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems. Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains, Lecture Notes in Mathematics, 1991. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-12245-3.

[14]

E. M. Harrell and J. Stubbe, On trace identities and universal eigenvalue estimates for some partial differential operators, Trans. Amer. Math. Soc., 349 (1997), 1797-1809.  doi: 10.1090/S0002-9947-97-01846-1.

[15]

B. Kawohl, H. A. Levine and W. Velte, Buckling eigenvalues for a clamped plate embedded in an elastic medium and related questions, SIAM J. Math. Anal., 24 (1993), 327–340. doi: 10.1137/0524022.

[16]

P. Kröger, Upper bounds for the Neumann eigenvalues on a bounded domain in Euclidean space, J. Funct. Anal., 106 (1992), 353–357. doi: 10.1016/0022-1236(92)90052-K.

[17]

P. Kröger, Estimates for sums of eigenvalues of the Laplacian, J. Funct. Anal., 126 (1994), 217–227. doi: 10.1006/jfan.1994.1146.

[18]

A. Laptev, Dirichlet and Neumann eigenvalue problems on domains in Euclidean spaces, J. Funct. Anal., 151 (1997), 531–545. doi: 10.1006/jfan.1997.3155.

[19]

R. S. Laugesen and B. A. Siudeja, Sums of Laplace eigenvalues–rotationally symmetric maximizers in the plane, J. Funct. Anal., 260 (2011), 1795–1823. doi: 10.1016/j.jfa.2010.12.018.

[20]

L. Li and L. Tang, Some upper bounds for sums of eigenvalues of the Neumann Laplacian, Proc. Amer. Math. Soc., 134 (2006), 3301–3307. doi: 10.1090/S0002-9939-06-08355-9.

[21]

P. Li and S. T. Yau, On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys., 88 (1983), 309–318.

[22]

A. D. Melas, A lower bound for sums of eigenvalues of the Laplacian, Proc. Amer. Math. Soc., 131 (2003), 631–636. doi: 10.1090/S0002-9939-02-06834-X.

[23]

N. Nadirashvili, Rayleigh's conjecture on the principal frequency of the clamped plate, Arch. Ration. Mech. Anal., 129 (1995), 1-10.  doi: 10.1007/BF00375124.

[24]

Y. Safarov and D. Vassiliev, The Asymptotic Distribution of Eigenvalues of Partial Differential Operators (English summary), Translated from the Russian manuscript by the authors. Translations of Mathematical Monographs, 155. American Mathematical Society, Providence, RI, 1997.

[25]

B. A. Siudeja, Generalized tight $p$-frames and spectral bounds for Laplacian-like operators, Appl. Comput. Harmon. Anal., 42 (2017), 167-198.  doi: 10.1016/j.acha.2015.08.001.

[26]

G. Szegő, Inequalities for certain eigenvalues of a membrane of given area, J. Rational Mech. Anal., 3 (1954), 343-356.  doi: 10.1512/iumj.1954.3.53017.

[27]

G. Talenti, On the first eigenvalue of the clamped plate, Ann. Mat. Pura Appl., 129 (1981), 265-280.  doi: 10.1007/BF01762146.

[28]

S. Yıldırım Yolcu and T. Yolcu, Estimates on the eigenvalues of the clamped plate problem on domains in Euclidean spaces, J. Math. Phys., 54 (2013), 043515, 13 pp. doi: 10.1063/1.4801446.

[29]

H. Weinberger, An isoperimetric inequality for the $N$-dimensional free membrane problem, J. Rational Mech. Anal., 5 (1956), 633-636.  doi: 10.1512/iumj.1956.5.55021.

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