Payne-Sperb-Stakgold Type Inequality for a Wedge-Like Membrane

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Abir Sboui
Abdelhalim Hasnaoui

Abstract

For a bounded domain contained in a wedge, we give a new Payne-Sperb-Stakgold type inequality for the solution of a semi-linear equation. The result is isoperimetric in the sense that the sector is the unique extremal domain.

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References

  1. G. Chiti, An Isoperimetric Inequality for the Eigenfunctions of Linear Second Order Elliptic Operators, Boll. Unione Mat. Ital., VI. Ser., A. 1 (1982), 145-151.
  2. A. Hasnaoui, L. Hermi, Isoperimetric Inequalities for a Wedge-Like Membrane, Ann. Henri Poincaré. 15 (2014), 369–406. https://doi.org/10.1007/s00023-013-0243-y.
  3. S. Kesavan, Topics in Functional Analysis and Applications, New Age International, 1989.
  4. S. Kesavan, P. Filomena, Symmetry of Positive Solutions of a Quasilinear Elliptic Equation via Isoperimetric Inequalities, Appl. Anal. 54 (1994), 27–37. https://doi.org/10.1080/00036819408840266.
  5. M.-T. Kohler-Jobin, Sur la première fonction propre d’une membrane: une extension àN dimensions de l’inégalité isopérimétrique de Payne-Rayner, J. Appl. Math. Phys. (ZAMP). 28 (1977), 1137–1140. https://doi.org/10.1007/BF01601680.
  6. M.-T. Kohler-Jobin, Isoperimetric Monotonicity and Isoperimetric Inequalities of Payne-Rayner Type for the First Eigenfunction of the Helmholtz Problem, Z. Angew. Math. Phys. 32 (1981), 625–646. https://doi.org/10.1007/BF00946975.
  7. J. Mossino, A Generalization of the Payne-Rayner Isoperimetric Inequality, Boll. Unione Mat. Ital., VI. Ser., A. 2 (1983), 335–342.
  8. 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, 27 (1962), 1-58.
  9. 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.
  10. L.E. Payne, M.E. Rayner, Some Isoperimetric Norm Bounds for Solutions of the Helmholtz Equation, J. Appl. Math. Phys. (ZAMP). 24 (1973), 105–110. https://doi.org/10.1007/BF01594001.
  11. L.E. Payne, R. Sperb, I. Stakgold, On Hopf Type Maximum Principles for Convex Domains, Nonlinear Anal., Theory Methods Appl. 1 (1977), 547–559. https://doi.org/10.1016/0362-546X(77)90016-5.
  12. 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.