• Previous Article
    Layered solutions to the vector Allen-Cahn equation in $\mathbb{R}^2$. Minimizers and heteroclinic connections
  • CPAA Home
  • This Issue
  • Next Article
    Global well-posedness of the two-dimensional horizontally filtered simplified Bardina turbulence model on a strip-like region
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.

[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.

[1]

Erwann Delay, Pieralberto Sicbaldi. Extremal domains for the first eigenvalue in a general compact Riemannian manifold. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5799-5825. doi: 10.3934/dcds.2015.35.5799

[2]

Julián Fernández Bonder, Leandro M. Del Pezzo. An optimization problem for the first eigenvalue of the $p-$Laplacian plus a potential. Communications on Pure & Applied Analysis, 2006, 5 (4) : 675-690. doi: 10.3934/cpaa.2006.5.675

[3]

Jan Sokołowski, Jan Stebel. Shape optimization for non-Newtonian fluids in time-dependent domains. Evolution Equations & Control Theory, 2014, 3 (2) : 331-348. doi: 10.3934/eect.2014.3.331

[4]

Luigi Ambrosio, Michele Miranda jr., Diego Pallara. Sets with finite perimeter in Wiener spaces, perimeter measure and boundary rectifiability. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 591-606. doi: 10.3934/dcds.2010.28.591

[5]

Wei-Ming Ni, Xuefeng Wang. On the first positive Neumann eigenvalue. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 1-19. doi: 10.3934/dcds.2007.17.1

[6]

Robert Brooks and Eran Makover. The first eigenvalue of a Riemann surface. Electronic Research Announcements, 1999, 5: 76-81.

[7]

Valeria Chiado Piat, Sergey S. Nazarov, Andrey Piatnitski. Steklov problems in perforated domains with a coefficient of indefinite sign. Networks & Heterogeneous Media, 2012, 7 (1) : 151-178. doi: 10.3934/nhm.2012.7.151

[8]

Barbara Kaltenbacher, Gunther Peichl. The shape derivative for an optimization problem in lithotripsy. Evolution Equations & Control Theory, 2016, 5 (3) : 399-430. doi: 10.3934/eect.2016011

[9]

Wenya Ma, Yihang Hao, Xiangao Liu. Shape optimization in compressible liquid crystals. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1623-1639. doi: 10.3934/cpaa.2015.14.1623

[10]

Shenggui Zhang. A sufficient condition of Euclidean rings given by polynomial optimization over a box. Numerical Algebra, Control & Optimization, 2014, 4 (2) : 93-101. doi: 10.3934/naco.2014.4.93

[11]

Eugenia Pérez. On periodic Steklov type eigenvalue problems on half-bands and the spectral homogenization problem. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 859-883. doi: 10.3934/dcdsb.2007.7.859

[12]

Vincenzo Ferone, Carlo Nitsch, Cristina Trombetti. On a conjectured reverse Faber-Krahn inequality for a Steklov--type Laplacian eigenvalue. Communications on Pure & Applied Analysis, 2015, 14 (1) : 63-82. doi: 10.3934/cpaa.2015.14.63

[13]

Murat Adivar, Shu-Cherng Fang. Convex optimization on mixed domains. Journal of Industrial & Management Optimization, 2012, 8 (1) : 189-227. doi: 10.3934/jimo.2012.8.189

[14]

Benedict Geihe, Martin Rumpf. A posteriori error estimates for sequential laminates in shape optimization. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1377-1392. doi: 10.3934/dcdss.2016055

[15]

Günter Leugering, Jan Sokołowski, Antoni Żochowski. Control of crack propagation by shape-topological optimization. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2625-2657. doi: 10.3934/dcds.2015.35.2625

[16]

Markus Muhr, Vanja Nikolić, Barbara Wohlmuth, Linus Wunderlich. Isogeometric shape optimization for nonlinear ultrasound focusing. Evolution Equations & Control Theory, 2019, 8 (1) : 163-202. doi: 10.3934/eect.2019010

[17]

Gang Meng. The optimal upper bound for the first eigenvalue of the fourth order equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6369-6382. doi: 10.3934/dcds.2017276

[18]

Yufei Sun, Grace Aw, Kok Lay Teo, Guanglu Zhou. Portfolio optimization using a new probabilistic risk measure. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1275-1283. doi: 10.3934/jimo.2015.11.1275

[19]

Craig Cowan, Pierpaolo Esposito, Nassif Ghoussoub. Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1033-1050. doi: 10.3934/dcds.2010.28.1033

[20]

Chiu-Yen Kao, Yuan Lou, Eiji Yanagida. Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains. Mathematical Biosciences & Engineering, 2008, 5 (2) : 315-335. doi: 10.3934/mbe.2008.5.315

2017 Impact Factor: 0.884

Metrics

  • PDF downloads (5)
  • HTML views (16)
  • Cited by (0)

Other articles
by authors

[Back to Top]