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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Adolfsson V.: L 2-integrability of second order derivatives for Poisson’s equation in nonsmooth domains. Math. Scand. 70, 146–160 (1992)
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)
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)
Berchio E., Gazzola F., Mitidieri E.: Positivity preserving property for a class of biharmonic elliptic problems. J. Differ. Equa. 229, 1–23 (2006)
Berchio E., Gazzola F., Weth T.: Critical growth biharmonic elliptic problems under Steklov-type boundary conditions. Adv. Differ. Equa. 12, 381–406 (2007)
Bucur D., Buttazzo G.: Variational methods in shape optimization problems. Progress in nonlinear differential equations and their applications, vol. 65. Birkhäuser Boston, Boston (2005)
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)
Ferrero A., Gazzola F., Weth T.: On a fourth order Steklov eigenvalue problem. Analysis 25, 315–332 (2005)
Fichera G.: Su un principio di dualità per talune formole di maggiorazione relative alle equazioni differenziali. Atti Accad. Naz. Lincei 19, 411–418 (1955)
Gazzola F., Sweers G.: On positivity for the biharmonic operator under Steklov boundary conditions. Arch. Rat. Mech. Anal. 188, 399–427 (2008)
Jerison D.S., Kenig C.E.: The Neumann problem on Lipschitz domains. Bull. Am. Math. Soc. 4, 203–207 (1981)
Jerison D.S., Kenig C.E.: Boundary value problems on Lipschitz domains. Stud. Part. Differ. Equa. 23, 1–68 (1982)
Krahn E.: Über eine von Rayleigh formulierte minimaleigenschaft des kreises. Math. Ann. 94, 97–100 (1925)
Krahn E.: Über minimaleigenschaften der kugel in drei und mehr dimensionen. Acta Commun. Univ. Dorpat. A9, 1–44 (1926)
Kuttler J.R.: Remarks on a Stekloff eigenvalue problem. SIAM J. Numer. Anal. 9, 1–5 (1972)
Kuttler J.R.: Dirichlet eigenvalues. SIAM J. Numer. Anal. 16, 332–338 (1979)
Kuttler J.R., Sigillito V.G.: Inequalities for membrane and Stekloff eigenvalues. J. Math. Anal. Appl. 23, 148–160 (1968)
Kuttler, J.R., Sigillito, V.G.: Estimating eigenvalues with a posteriori/a priori inequalities. Research Notes in Mathematics, Pitman Advanced Publishing Program (1985)
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)
Necas, J.: Les méthodes directes en théorie des équations elliptiques, Masson et C ie Editeurs, Paris (1967)
Payne L.E.: Some isoperimetric inequalities for harmonic functions. SIAM J. Math. Anal. 1, 354–359 (1970)
Simon J.: Differentiation with respect to the domain in boundary value problems. Numer. Funct. Anal. Optim. 2, 649–687 (1980)
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)
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)
Stekloff W.: Sur les problèmes fondamentaux de la physique mathématique. Ann. Sci. Ecol. Norm. Sup. 19, 455–490 (1902)
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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