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

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Abir Sboui
Abdelhalim Hasnaoui


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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  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.
  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.
  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.
  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.
  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.
  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.
  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.
  12. L.E. Payne, H.F. Weinberger, A Faber-Krahn Inequality for Wedge-Like Membranes, J. Math. Phys. 39 (1960), 182–188.