Chiti-type Reverse Hölder Inequality and Saint-Venant Theorem for Wedge Domains on Spheres

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-22-2024-87
Published: May 27, 2024
Cite as:
Abdelhalim Hasnaoui, Abdelhamid Zaghdani, Chiti-type Reverse Hölder Inequality and Saint-Venant Theorem for Wedge Domains on Spheres, Int. J. Anal. Appl., 22 (2024), 87.

Main Article Content

Abdelhalim Hasnaoui, Abdelhamid Zaghdani

Abstract

In this paper, we prove a new weighted reverse Hölder inequality for the first eigenfunction of the Dirichlet eigenvalue problem in a domain completely contained in a wedge in the sphere S2. This inequality is known as the Payne-Rayner inequality or Chiti-type inequality. We also prove an extension of Saint-Venant inequality for the relative torsional rigidity of such domains.

Article Details

References

  1. L.E. Payne, H.F. Weinberger, A Faber-Krahn Inequality for Wedge-Like Membranes, J. Math. Phys. 39 (1960), 182–188. https://doi.org/10.1002/sapm1960391182.
  2. C. Bandle, Isoperimetric Inequalities and Applications, Monographs and Studies in Mathematics, Vol. 7, Pitman, Boston, (1980).
  3. L.E. Payne, Isoperimetric Inequalities for Eigenvalue and Their Applications, Autovalori e Autosoluzioni: Lectures Given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.) Held in Chieti, Italy, 1962, G. Fichera (ed.), C.I.M.E. Summer Schools, Vol. 27, 1–58, (1962).
  4. A. Hasnaoui, L. Hermi, Isoperimetric Inequalities for a Wedge-Like Membrane, Ann. Henri Poinc. 15 (2013), 369–406. https://doi.org/10.1007/s00023-013-0243-y.
  5. L.E. Payne, M.E. Rayner, An Isoperimetric Inequality for the First Eigenfunction in the Fixed Membrane Problem, J. Appl. Math. Phys. (ZAMP) 23 (1972), 13–15. https://doi.org/10.1007/bf01593198.
  6. G. Chiti, A Reverse Hölder Inequality for the Eigenfunctions of Linear Second Order Elliptic Operators, Z. Angew. Math. Phys. 33 (1982), 143–148. https://doi.org/10.1007/BF00948319.
  7. A. Weinstein, Generalized Axially Symmetric Potential Theory, Bull. Amer. Math. Soc. 59 (1953), 20–38. https://doi.org/10.1090/s0002-9904-1953-09651-3.
  8. J. Ratzkin, A. Treibergs, A Payne-Weinberger Eigenvalue Estimate for Wedge Domains on Spheres, Proc. Amer. Math. Soc. 137 (2009), 2299–2309. https://doi.org/10.1090/s0002-9939-09-09790-1.
  9. G.A. Philippin, Some Isoperimetric Norm Bounds for the First Eigenfunction of Wedge-Like Membranes, Z. Angew. Math. Phys. 27 (1976), 545–551. https://doi.org/10.1007/BF01591165.
  10. A. Hasnaoui, L. Hermi, A Sharp Upper Bound for the First Dirichlet Eigenvalue of a Class of Wedge-Like Domains, Z. Angew. Math. Phys. 66 (2015), 2419–2440. https://doi.org/10.1007/s00033-015-0530-1.
  11. G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities, Second Edition, Cambridge University Press, Cambridge, (1952).
  12. F. Brock, F. Chiacchio, A. Mercaldo, Weighted Isoperimetric Inequalities in Cones and Applications, Nonlinear Anal.: Theory Meth. Appl. 75 (2012), 5737–5755. https://doi.org/10.1016/j.na.2012.05.011.
  13. F. Brock, F. Chiacchio, A. Mercaldo, A Class of Degenerate Elliptic Equations and a Dido’s Problem With Respect to a Measure, J. Math. Anal. Appl. 348 (2008), 356–365. https://doi.org/10.1016/j.jmaa.2008.07.010.
  14. C. Maderna, S. Salsa, Sharp Estimates for Solutions to a Certain Type of Singular Elliptic Boundary Value Problems in Two Dimensions, Appl. Anal. 12 (1981), 307–321. https://doi.org/10.1080/00036818108839370.