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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.

[2]

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

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[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.

[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.

[17]

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

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.

[2]

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

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[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.

[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.

[17]

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

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