Proof of the Payne-pólya-weinberger Conjecture

@inproceedings{ASHBAUGH2007ProofOT,
  title={Proof of the Payne-pólya-weinberger Conjecture},
  author={MARK S. ASHBAUGH and Rafael D. Benguria},
  year={2007}
}
In 1955 and 1956 Payne, Pólya, and Weinberger considered the problem of bounding ratios of eigenvalues for homogeneous membranes of arbitrary shape [PPW1, PPW2]. Among other things, they showed that the ratio k2IK °f * ^ r s t t w o eigenvalues was less than or equal to 3 and went on to conjecture that the optimal upper bound for A2/Aj was its value for the disk, approximately 2.539. It is this conjecture which we establish below. Since 1956 various authors have attempted to prove the… CONTINUE READING

From This Paper

Topics from this paper.

References

Publications referenced by this paper.
Showing 1-9 of 9 references

Rearrangements and convexity of level sets in PDE

  • B. Kawohl
  • Lecture Notes in Math., vol. 1150, Springer…
  • 1985

A reverse Holder inequality for the eigenfunctions of linear second order elliptic operators

  • G. Chiti
  • J. Appl. Math, and Phys. (ZAMP)
  • 1982

Isoperimetric inequalities and applications

  • C. Bandle
  • Pitman, Boston
  • 1980
1 Excerpt

Luttinger, A general rearrangement inequality for multiple integrals

  • H. J. Brascamp, E. H. Lieb, M J.
  • J. Funct. Anal
  • 1974

On the upper bound for the ratio of the first two membrane eigenvalues

  • H. L. de Vries
  • Z. Natur
  • 1967

On symmetric membranes and conformai radius: Some complements to Pólya's and Szegb's inequalities, Arch

  • J. Hersch
  • Rational Mech. Anal
  • 1965

An isoperimetric inequality for the N-dimensional free membrane

  • H. F. Weinberger
  • problem, J. Rational Mech. Anal
  • 1956

Sur le quotient de deux fréquences propres consécutives

  • PPWl L.E. Payne, G. Pólya, H. F. Weinberger
  • C. R. Acad. Sci. Paris
  • 1955

Inequalities

  • G. H. Hardy, J. E. Littlewood, G. Pólya
  • 2nd éd., Cambridge Univ. Press, Cambridge
  • 1952

Similar Papers

Loading similar papers…