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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[8]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[8]

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

[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.  Google Scholar

[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.  Google Scholar

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