• Previous Article
    Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponents
  • CPAA Home
  • This Issue
  • Next Article
    Qualitative properties of stationary solutions of the NLS on the Hyperbolic space without and with external potentials
September  2019, 18(5): 2679-2691. doi: 10.3934/cpaa.2019119

Faber-Krahn and Lieb-type inequalities for the composite membrane problem

1. 

Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy

2. 

Dipartimento di Matematica "Guido Castelnuovo", Sapienza Università di Roma, P.le Aldo Moro 5, 00185 Roma, Italy

* Corresponding author

Received  June 2018 Revised  October 2018 Published  April 2019

The classical Faber-Krahn inequality states that, among all domains with given measure, the ball has the smallest first Dirichlet eigenvalue of the Laplacian. Another inequality related to the first eigenvalue of the Laplacian has been proved by Lieb in 1983 and it relates the first Dirichlet eigenvalues of the Laplacian of two different domains with the first Dirichlet eigenvalue of the intersection of translations of them. In this paper we prove the analogue of Faber-Krahn and Lieb inequalities for the composite membrane problem.

Citation: Giovanni Cupini, Eugenio Vecchi. Faber-Krahn and Lieb-type inequalities for the composite membrane problem. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2679-2691. doi: 10.3934/cpaa.2019119
References:
[1]

A. AlvinoP. L. Lions and G. Trombetti, A remark on comparison results via symmetrization, Proc. Roy. Soc. Edinburgh Sect A, 102 (1986), 37-49.  doi: 10.1017/S0308210500014475.  Google Scholar

[2]

I. Birindelli, G. Galise and H. Ishii, Towards a reversed Faber-Krahn inequality for the truncated Laplacian, preprint, arXiv: 1803.07362. Google Scholar

[3]

S. ChanilloD. GrieserM. ImaiK. Kurata and I. Ohnishi, Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes, Comm. Math. Phys., 214 (2000), 315-337.  doi: 10.1007/PL00005534.  Google Scholar

[4]

S. Chanillo, D. Grieser and K. Kurata, The free boundary problem in the optimization of composite membranes, in Differential Geometric Methods in The Control of Partial Differential Equations (Boulder, CO, 1999), Amer. Math. Soc., Providence, RI, 2000, 268, 61–81. doi: 10.1090/conm/268/04308.  Google Scholar

[5]

S. Chanillo and C. E. Kenig, Weak uniqueness and partial regularity for the composite membrane problem, J. Eur. Math. Soc., 10 (2008), 705-737.  doi: 10.4171/JEMS/127.  Google Scholar

[6]

S. ChanilloC. E. Kenig and T. To, Regularity of the minimizers in the composite membrane problem in $\mathbb{R}^2$, J. Funct. Anal., 255 (2008), 2299-2320.  doi: 10.1016/j.jfa.2008.04.015.  Google Scholar

[7]

F. Colasuonno and E. Vecchi, Symmetry in the composite plate problem, Commun. Contemp. Math., (2018).  doi: 10.1142/S0219199718500190.  Google Scholar

[8]

F. Colasuonno and E. Vecchi, Symmetry and rigidity for the hinged composite plate problem, J. Differential Equations, (2018).  doi: 10.1016/j.jde.2018.10.011.  Google Scholar

[9]

G. Faber, Beweiss dass unter alien homogenen Membranen von gleicher Flache und gleicher Spannung die kreisformgige den leifsten Grundton gibt, Sitz. bayer Acad. Wiss., (1923), 169-172.   Google Scholar

[10]

A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Birkhäuser Verlag, Basel, 2006.  Google Scholar

[11]

B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Springer-Verlag, Berlin, 1985. doi: 10.1007/BFb0075060.  Google Scholar

[12]

S. Kesavan, Some remarks on a result of Talenti, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 15 (1988), 453-465.   Google Scholar

[13]

S. Kesavan, Symmetrization and Applications, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. doi: 10.1142/9789812773937.  Google Scholar

[14]

E. Krahn, Über eine von Rayleigh formulierte Minimaleigenschaftdes Kreises, Math. Ann., 94 (1924), 97-100.  doi: 10.1007/BF01208645.  Google Scholar

[15]

E. H. Lieb, On the lowest eigenvalue of the Laplacian for the intersection of two domains, Invent. Math., 74 (1983), 441-448.  doi: 10.1007/BF01394245.  Google Scholar

[16]

H. Shahgholian, The singular set for the composite membrane problem, Comm. Math. Phys., 271 (2007), 93-101.  doi: 10.1007/s00220-006-0160-8.  Google Scholar

[17]

G. Talenti, Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 3 (1976), 697-718.   Google Scholar

show all references

References:
[1]

A. AlvinoP. L. Lions and G. Trombetti, A remark on comparison results via symmetrization, Proc. Roy. Soc. Edinburgh Sect A, 102 (1986), 37-49.  doi: 10.1017/S0308210500014475.  Google Scholar

[2]

I. Birindelli, G. Galise and H. Ishii, Towards a reversed Faber-Krahn inequality for the truncated Laplacian, preprint, arXiv: 1803.07362. Google Scholar

[3]

S. ChanilloD. GrieserM. ImaiK. Kurata and I. Ohnishi, Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes, Comm. Math. Phys., 214 (2000), 315-337.  doi: 10.1007/PL00005534.  Google Scholar

[4]

S. Chanillo, D. Grieser and K. Kurata, The free boundary problem in the optimization of composite membranes, in Differential Geometric Methods in The Control of Partial Differential Equations (Boulder, CO, 1999), Amer. Math. Soc., Providence, RI, 2000, 268, 61–81. doi: 10.1090/conm/268/04308.  Google Scholar

[5]

S. Chanillo and C. E. Kenig, Weak uniqueness and partial regularity for the composite membrane problem, J. Eur. Math. Soc., 10 (2008), 705-737.  doi: 10.4171/JEMS/127.  Google Scholar

[6]

S. ChanilloC. E. Kenig and T. To, Regularity of the minimizers in the composite membrane problem in $\mathbb{R}^2$, J. Funct. Anal., 255 (2008), 2299-2320.  doi: 10.1016/j.jfa.2008.04.015.  Google Scholar

[7]

F. Colasuonno and E. Vecchi, Symmetry in the composite plate problem, Commun. Contemp. Math., (2018).  doi: 10.1142/S0219199718500190.  Google Scholar

[8]

F. Colasuonno and E. Vecchi, Symmetry and rigidity for the hinged composite plate problem, J. Differential Equations, (2018).  doi: 10.1016/j.jde.2018.10.011.  Google Scholar

[9]

G. Faber, Beweiss dass unter alien homogenen Membranen von gleicher Flache und gleicher Spannung die kreisformgige den leifsten Grundton gibt, Sitz. bayer Acad. Wiss., (1923), 169-172.   Google Scholar

[10]

A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Birkhäuser Verlag, Basel, 2006.  Google Scholar

[11]

B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Springer-Verlag, Berlin, 1985. doi: 10.1007/BFb0075060.  Google Scholar

[12]

S. Kesavan, Some remarks on a result of Talenti, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 15 (1988), 453-465.   Google Scholar

[13]

S. Kesavan, Symmetrization and Applications, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. doi: 10.1142/9789812773937.  Google Scholar

[14]

E. Krahn, Über eine von Rayleigh formulierte Minimaleigenschaftdes Kreises, Math. Ann., 94 (1924), 97-100.  doi: 10.1007/BF01208645.  Google Scholar

[15]

E. H. Lieb, On the lowest eigenvalue of the Laplacian for the intersection of two domains, Invent. Math., 74 (1983), 441-448.  doi: 10.1007/BF01394245.  Google Scholar

[16]

H. Shahgholian, The singular set for the composite membrane problem, Comm. Math. Phys., 271 (2007), 93-101.  doi: 10.1007/s00220-006-0160-8.  Google Scholar

[17]

G. Talenti, Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 3 (1976), 697-718.   Google Scholar

[1]

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

[2]

Janusz Mierczyński, Wenxian Shen. The Faber--Krahn inequality for random/nonautonomous parabolic equations. Communications on Pure & Applied Analysis, 2005, 4 (1) : 101-114. doi: 10.3934/cpaa.2005.4.101

[3]

Neal Bez, Sanghyuk Lee, Shohei Nakamura, Yoshihiro Sawano. Sharpness of the Brascamp–Lieb inequality in Lorentz spaces. Electronic Research Announcements, 2017, 24: 53-63. doi: 10.3934/era.2017.24.006

[4]

S. J. Li, Z. M. Fang. On the stability of a dual weak vector variational inequality problem. Journal of Industrial & Management Optimization, 2008, 4 (1) : 155-165. doi: 10.3934/jimo.2008.4.155

[5]

Wenhui Shi. An epiperimetric inequality approach to the parabolic Signorini problem. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1813-1846. doi: 10.3934/dcds.2020095

[6]

Stanislaw Migórski. Hemivariational inequality for a frictional contact problem in elasto-piezoelectricity. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1339-1356. doi: 10.3934/dcdsb.2006.6.1339

[7]

T. A. Shaposhnikova, M. N. Zubova. Homogenization problem for a parabolic variational inequality with constraints on subsets situated on the boundary of the domain. Networks & Heterogeneous Media, 2008, 3 (3) : 675-689. doi: 10.3934/nhm.2008.3.675

[8]

Junfeng Yang. Dynamic power price problem: An inverse variational inequality approach. Journal of Industrial & Management Optimization, 2008, 4 (4) : 673-684. doi: 10.3934/jimo.2008.4.673

[9]

Stanisław Migórski. A note on optimal control problem for a hemivariational inequality modeling fluid flow. Conference Publications, 2013, 2013 (special) : 545-554. doi: 10.3934/proc.2013.2013.545

[10]

Zhenhai Liu, Stanislaw Migórski. Noncoercive damping in dynamic hemivariational inequality with application to problem of piezoelectricity. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 129-143. doi: 10.3934/dcdsb.2008.9.129

[11]

Jianlin Jiang, Shun Zhang, Su Zhang, Jie Wen. A variational inequality approach for constrained multifacility Weber problem under gauge. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1085-1104. doi: 10.3934/jimo.2017091

[12]

Leszek Gasiński. Optimal control problem of Bolza-type for evolution hemivariational inequality. Conference Publications, 2003, 2003 (Special) : 320-326. doi: 10.3934/proc.2003.2003.320

[13]

Yekini Shehu, Olaniyi Iyiola. On a modified extragradient method for variational inequality problem with application to industrial electricity production. Journal of Industrial & Management Optimization, 2019, 15 (1) : 319-342. doi: 10.3934/jimo.2018045

[14]

José Francisco de Oliveira, João Marcos do Ó, Pedro Ubilla. Hardy-Sobolev type inequality and supercritical extremal problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3345-3364. doi: 10.3934/dcds.2019138

[15]

Xiaona Fan, Li Jiang, Mengsi Li. Homotopy method for solving generalized Nash equilibrium problem with equality and inequality constraints. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1795-1807. doi: 10.3934/jimo.2018123

[16]

Jorge A. Becerril, Javier F. Rosenblueth. Necessity for isoperimetric inequality constraints. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1129-1158. doi: 10.3934/dcds.2017047

[17]

Gisella Croce, Bernard Dacorogna. On a generalized Wirtinger inequality. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1329-1341. doi: 10.3934/dcds.2003.9.1329

[18]

Changjun Yu, Kok Lay Teo, Liansheng Zhang, Yanqin Bai. On a refinement of the convergence analysis for the new exact penalty function method for continuous inequality constrained optimization problem. Journal of Industrial & Management Optimization, 2012, 8 (2) : 485-491. doi: 10.3934/jimo.2012.8.485

[19]

YanYan Li, Tonia Ricciardi. A sharp Sobolev inequality on Riemannian manifolds. Communications on Pure & Applied Analysis, 2003, 2 (1) : 1-31. doi: 10.3934/cpaa.2003.2.1

[20]

Felipe Riquelme. Ruelle's inequality in negative curvature. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2809-2825. doi: 10.3934/dcds.2018119

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (23)
  • HTML views (116)
  • Cited by (0)

Other articles
by authors

[Back to Top]