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On the first eigenvalue of a fourth order Steklov problem

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Abstract

We prove some results about the first Steklov eigenvalue d 1 of the biharmonic operator in bounded domains. Firstly, we show that Fichera’s principle of duality (Fichera in Atti Accad Naz Lincei 19:411–418, 1955) may be extended to a wide class of nonsmooth domains. Next, we study the optimization of d 1 for varying domains: we disprove a long-standing conjecture, we show some new and unexpected features and we suggest some challenging problems. Finally, we prove several properties of the ball.

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References

  1. Adolfsson V.: L 2-integrability of second order derivatives for Poisson’s equation in nonsmooth domains. Math. Scand. 70, 146–160 (1992)

    MATH  MathSciNet  Google Scholar 

  2. Agmon S., Douglis A., Nirenberg L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I. Commun. Pure Appl. Math. 12, 623–727 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  3. Babuška, I.: Stabilität des Definitionsgebietes mit Rücksicht auf grundlegende Probleme der Theorie der partiellen Differentialgleichungen auch im Zusammenhang mit der Elastizitätstheorie. I, II. Czech. Math. J. 11(86), 76–105, 165–203 (1961)

  4. Berchio E., Gazzola F., Mitidieri E.: Positivity preserving property for a class of biharmonic elliptic problems. J. Differ. Equa. 229, 1–23 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Berchio E., Gazzola F., Weth T.: Critical growth biharmonic elliptic problems under Steklov-type boundary conditions. Adv. Differ. Equa. 12, 381–406 (2007)

    MATH  MathSciNet  Google Scholar 

  6. Bucur D., Buttazzo G.: Variational methods in shape optimization problems. Progress in nonlinear differential equations and their applications, vol. 65. Birkhäuser Boston, Boston (2005)

    Google Scholar 

  7. Faber, G.: Beweis, dass unter allen homogenen membranen von gleicher fläche und gleicher spannung die kreisförmige den tiefsten grundton gibt. Sitz. Ber. Bayer. Akad. Wiss. 169–172 (1923)

  8. Ferrero A., Gazzola F., Weth T.: On a fourth order Steklov eigenvalue problem. Analysis 25, 315–332 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Fichera G.: Su un principio di dualità per talune formole di maggiorazione relative alle equazioni differenziali. Atti Accad. Naz. Lincei 19, 411–418 (1955)

    MathSciNet  Google Scholar 

  10. Gazzola F., Sweers G.: On positivity for the biharmonic operator under Steklov boundary conditions. Arch. Rat. Mech. Anal. 188, 399–427 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. Jerison D.S., Kenig C.E.: The Neumann problem on Lipschitz domains. Bull. Am. Math. Soc. 4, 203–207 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  12. Jerison D.S., Kenig C.E.: Boundary value problems on Lipschitz domains. Stud. Part. Differ. Equa. 23, 1–68 (1982)

    MathSciNet  Google Scholar 

  13. Krahn E.: Über eine von Rayleigh formulierte minimaleigenschaft des kreises. Math. Ann. 94, 97–100 (1925)

    Article  MATH  MathSciNet  Google Scholar 

  14. Krahn E.: Über minimaleigenschaften der kugel in drei und mehr dimensionen. Acta Commun. Univ. Dorpat. A9, 1–44 (1926)

    Google Scholar 

  15. Kuttler J.R.: Remarks on a Stekloff eigenvalue problem. SIAM J. Numer. Anal. 9, 1–5 (1972)

    Article  MathSciNet  Google Scholar 

  16. Kuttler J.R.: Dirichlet eigenvalues. SIAM J. Numer. Anal. 16, 332–338 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kuttler J.R., Sigillito V.G.: Inequalities for membrane and Stekloff eigenvalues. J. Math. Anal. Appl. 23, 148–160 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kuttler, J.R., Sigillito, V.G.: Estimating eigenvalues with a posteriori/a priori inequalities. Research Notes in Mathematics, Pitman Advanced Publishing Program (1985)

  19. Nazarov S.A., Sweers G.: A hinged plate equation and iterated Dirichlet Laplace operator on domains with concave corners. J. Diff. Eq. 233, 151–180 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  20. Necas, J.: Les méthodes directes en théorie des équations elliptiques, Masson et C ie Editeurs, Paris (1967)

  21. Payne L.E.: Some isoperimetric inequalities for harmonic functions. SIAM J. Math. Anal. 1, 354–359 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  22. Simon J.: Differentiation with respect to the domain in boundary value problems. Numer. Funct. Anal. Optim. 2, 649–687 (1980)

    MATH  MathSciNet  Google Scholar 

  23. Smith J.: The coupled equation approach to the numerical solution of the biharmonic equation by finite differences, I. SIAM J. Numer. Anal. 5, 323–339 (1968)

    Article  MathSciNet  Google Scholar 

  24. Smith J.: The coupled equation approach to the numerical solution of the biharmonic equation by finite differences, II. SIAM J. Numer. Anal. 7, 104–111 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  25. Stekloff W.: Sur les problèmes fondamentaux de la physique mathématique. Ann. Sci. Ecol. Norm. Sup. 19, 455–490 (1902)

    MathSciNet  Google Scholar 

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Correspondence to Filippo Gazzola.

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Bucur, D., Ferrero, A. & Gazzola, F. On the first eigenvalue of a fourth order Steklov problem. Calc. Var. 35, 103–131 (2009). https://doi.org/10.1007/s00526-008-0199-9

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  • DOI: https://doi.org/10.1007/s00526-008-0199-9

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