The First Biharmonic Steklov Eigenvalue: Positivity Preserving and Shape Optimization

Milan Journal of Mathematics - Tập 79 Số 1 - Trang 247-258 - 2011
Dorin Bucur1, Filippo Gazzola2
1Laboratoire de Mathématiques (LAMA) UMR 5127, Université de Savoie – Campus Scientifique, 73 376, Le-Bourget-Du-Lac, France
2Dipartimento di Matematica Politecnico di Milano, Milano, Italy

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Tài liệu tham khảo

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

Berchio E., Cassani D., Gazzola F.: Hardy-Rellich inequalities with boundary remainder terms and applications. Manuscripta Math. 131, 427–458 (2010)

Berchio E., Gazzola F., Mitidieri E.: Positivity preserving property for a class of biharmonic elliptic problems. J. Diff. Eq. 320, 1–23 (2006)

Brock F.: An isoperimetric inequality for eigenvalues of the Stekloff problem. Z. Angew. Math. Mech. 81, 69–71 (2001)

Bucur D., Ferrero A., Gazzola F.: On the first eigenvalue of a fourth order Steklov problem. Calc. Var. 35, 103–131 (2009)

Chenais D.: On the existence of a solution in a domain identification problem. J. Math. Anal. Appl. 52, 189–219 (1975)

Colesanti A., Fimiami M.: The Minkowski problem for the torsional rigidity. Indiana Univ. Math. J. 59, 1013–1039 (2010)

P. Destuynder, M. Salaun, Mathematical analysis of thin plate models. Mathématiques & Applications (Berlin) 24, Springer, 1996.

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

G. Fichera, Su un principio di dualità per talune formole di maggiorazione relative alle equazioni differenziali. Atti Acc. Naz. Lincei (8) 19 (1955), 411–418.

Friedrichs K.: Die randwert und eigenwertprobleme aus der theorie der elastischen platten. Math. Ann. 98, 205–247 (1927)

Fromm S.J.: Potential space estimates for Green potentials in convex domains. Proc. Amer. Math. Soc. 119, 225–233 (1993)

F. Gazzola, H.C. Grunau, G. Sweers, Polyharmonic boundary value problems. LNM 1991 Springer, 2010.

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

Jerison D.: A Minkowski problem for electrostatic capacity. Acta Math. 176, 1–47 (1996)

Kirchhoff G.R.: Über das gleichgewicht und die bewegung einer elastischen scheibe. J. Reine Angew. Math. 40, 51–88 (1850)

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)

Lakes R.S.: Foam structures with a negative Poisson’s ratio. Science 235, 1038–1040 (1987)

J.L. Lions, E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 3, Travaux et Recherches Mathématiques, No. 20, Dunod, 1970.

G. Liu, The Weyl-type asymptotic formula for biharmonic Steklov eigenvalues on Riemannian manifolds. Preprint

A.E.H. Love, A treatise on the mathematical theory of elasticity. 4th Edition, Cambridge Univ. Press, 1927.

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)

Parini E., Stylianou A.: On the positivity preserving property of hinged plates. SIAM J. Math. Anal. 41, 2031–2037 (2009)

L.E. Payne, Bounds for the maximum stress in the Saint Venant torsion problem. Indian J. Mech. Math. (1968/69), part I, 51–59. Special issue presented to Professor Bibhutibhusan Sen on the occasion of his seventieth birthday.

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

Philippin G.A., Safoui A.: On extending some maximum principles to convex domains with nonsmooth boundaries. Math. Methods Appl. Sci. 33, 1850–1855 (2010)

R. Schneider, Convex bodies: the Brunn-Minkowski theory. Cambridge Univ. Press, 1993

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)

W. Stekloff, Sur les problèmes fondamentaux de la physique mathématique. Ann. Sci. École Norm. Sup. (3) 19 (1902), 191–259 and 455–490.

Vitali G.: Sull’integrazione per serie. Rend. Circ. Mat. Palermo 23, 137–155 (1907)

Wang Q., Xia C.: Sharp bounds for the first non-zero Stekloff eigenvalues. J. Funct. Anal. 257, 2635–2644 (2009)