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September  2017, 16(5): 1843-1859. doi: 10.3934/cpaa.2017089

Optimality conditions of the first eigenvalue of a fourth order Steklov problem

Institut für Mathematik, RWTH Aachen, Templergraben 55, D-52062 Aachen, Germany

Received  November 2016 Revised  January 2017 Published  May 2017

In this paper we compute the first and second general domain variation of the first eigenvalue of a fourth order Steklov problem. We study optimality conditions for the ball among domains of given measure and among domains of given perimeter. We show that in both cases the ball is a local minimizer among all domains of equal measure and perimeter.

Citation: Monika Laskawy. Optimality conditions of the first eigenvalue of a fourth order Steklov problem. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1843-1859. doi: 10.3934/cpaa.2017089
References:
[1]

P. R. S. Antunes and F. Gazzola, Convex shape optimization for the least biharmonic Steklov eigenvalue, ESAIM: Control, Optimisation and Calculus of Variations, 19 (2013), 385-403.  doi: 10.1051/cocv/2012014.

[2]

C. Bandle and A. Wagner, Second domain variation for problems with robin boundary conditions, Journal of Optimization Theory and Applications, 2 (2015), 430-463.  doi: 10.1007/s10957-015-0801-1.

[3]

E. BerchioF. Gazzola and E. Mitidieri, Positivity preserving property for a class of biharmonic elliptic problems, J. Differential Equations, 229 (2006), 1-23.  doi: 10.1016/j.jde.2006.04.003.

[4]

D. BucurA. Ferrero and F. Gazzola, On the first eigenvalue of a fourth order Steklov problem, Calc. Var. Partial Differential Equations, 35 (2009), 103-131.  doi: 10.1007/s00526-008-0199-9.

[5]

D. Bucur and F. Gazzola, The first biharmonic Steklov eigenvalue: positivity preserving and shape optimization, Milan J. Math., 79 (2011), 247-258.  doi: 10.1007/s00032-011-0143-x.

[6]

A. FerreroF. Gazzola and T. Weth, On a fourth order Steklov eigenvalue problem, Analysis (Munich), 25 (2005), 315-332.  doi: 10.1524/anly.2005.25.4.315.

[7]

A. Henrot and M. Pierre, Variation et optimisation de formes, Springer, Berlin, 2005. doi: 10.1007/3-540-37689-5.

[8]

J. R. Kuttler, Remarks on a Stekloff eigenvalue problem, SIAM J. Numer. Anal., 9 (1972), 1-5.  doi: 10.1137/0709001.

[9]

G. Liu, The Weyl-type asymptotic formula for biharmonic Steklov eigenvalues on Riemannian manifolds, Adv. Math., 228 (2011), 2162-2217.  doi: 10.1016/j.aim.2011.07.001.

[10]

S. Raulot and A. Savo, Sharp bounds for the first eigenvalue of a fourth-order Steklov problem, J. Geom. Anal., 25 (2015), 1602-1619.  doi: 10.1007/s12220-014-9486-1.

show all references

References:
[1]

P. R. S. Antunes and F. Gazzola, Convex shape optimization for the least biharmonic Steklov eigenvalue, ESAIM: Control, Optimisation and Calculus of Variations, 19 (2013), 385-403.  doi: 10.1051/cocv/2012014.

[2]

C. Bandle and A. Wagner, Second domain variation for problems with robin boundary conditions, Journal of Optimization Theory and Applications, 2 (2015), 430-463.  doi: 10.1007/s10957-015-0801-1.

[3]

E. BerchioF. Gazzola and E. Mitidieri, Positivity preserving property for a class of biharmonic elliptic problems, J. Differential Equations, 229 (2006), 1-23.  doi: 10.1016/j.jde.2006.04.003.

[4]

D. BucurA. Ferrero and F. Gazzola, On the first eigenvalue of a fourth order Steklov problem, Calc. Var. Partial Differential Equations, 35 (2009), 103-131.  doi: 10.1007/s00526-008-0199-9.

[5]

D. Bucur and F. Gazzola, The first biharmonic Steklov eigenvalue: positivity preserving and shape optimization, Milan J. Math., 79 (2011), 247-258.  doi: 10.1007/s00032-011-0143-x.

[6]

A. FerreroF. Gazzola and T. Weth, On a fourth order Steklov eigenvalue problem, Analysis (Munich), 25 (2005), 315-332.  doi: 10.1524/anly.2005.25.4.315.

[7]

A. Henrot and M. Pierre, Variation et optimisation de formes, Springer, Berlin, 2005. doi: 10.1007/3-540-37689-5.

[8]

J. R. Kuttler, Remarks on a Stekloff eigenvalue problem, SIAM J. Numer. Anal., 9 (1972), 1-5.  doi: 10.1137/0709001.

[9]

G. Liu, The Weyl-type asymptotic formula for biharmonic Steklov eigenvalues on Riemannian manifolds, Adv. Math., 228 (2011), 2162-2217.  doi: 10.1016/j.aim.2011.07.001.

[10]

S. Raulot and A. Savo, Sharp bounds for the first eigenvalue of a fourth-order Steklov problem, J. Geom. Anal., 25 (2015), 1602-1619.  doi: 10.1007/s12220-014-9486-1.

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