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January  2021, 20(1): 145-158. doi: 10.3934/cpaa.2020261

A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle

1. 

Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Università degli studi di Napoli Federico Ⅱ, Via Cintia, Complesso Universitario Monte S. Angelo, 80126 Napoli, Italy

2. 

Dipartimento di Ingegneria Elettrica e dell'Informazione "M. Scarano", Università degli Studi di Cassino e del Lazio Meridionale, Via G. Di Biasio n. 43, 03043 Cassino (FR), Italy

* Corresponding author

Received  May 2020 Revised  August 2020 Published  January 2021 Early access  October 2020

Fund Project: This work has been partially supported by GNAMPA of INdAM and by Progetto di eccellenza "Sistemi distribuiti intelligenti" of Dipartimento di Ingegneria Elettrica e dell'Informazione "M. Scarano"

In this paper we study the first Steklov-Laplacian eigenvalue with an internal fixed spherical obstacle. We prove that the spherical shell locally maximizes the first eigenvalue among nearly spherical sets when both the internal ball and the volume are fixed.

Citation: Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure and Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261
References:
[1]

F. Brock, An isoperimetric inequality for eigenvalues of the Stekloff problem, Z. Angew. Math. Mech., 81 (2001), 69-71. 

[2]

D. Bucur, V. Ferone, C. Nitsch, C. Trombetti, Weinstock inequality in higher dimensions, J. Differential Geom. (in press). arXiv: 1710.04587

[3]

J. ChoeM. Ghomi and M. Ritoré, The relative isoperimetric inequality outside convex domains in $\mathbb{R}^n$, Calc. Var. Partial Differ. Equ., 29 (2007), 421-429.  doi: 10.1007/s00526-006-0027-z.

[4]

G. CrastaI. Fragalá and F. Gazzola, A sharp upper bound for the torsional rigidity of rods by means of web functions, Arch. Ration. Mech. Anal., 164 (2002), 189-211.  doi: 10.1007/s002050200205.

[5]

F. Della Pietra and G. Piscitelli, An optimal bound for nonlinear eigenvalues and torsional rigidity on domains with holes, Milan J. Math., (2020), 12pp. doi: 10.1007/s00032-020-00320-9.

[6]

M. Egert and P. Tolksdorf, Characterizations of Sobolev functions that vanish on a part of the boundary, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 729-743.  doi: 10.3934/dcdss.2017037.

[7]

V. FeroneC. Nitsch and C. Trombetti, On a conjectured reverse Faber-Krahn inequality for a Steklov-type Laplacian eigenvalue, Commun. Pure Appl. Anal., 14 (2015), 63-82.  doi: 10.3934/cpaa.2015.14.63.

[8]

I. Ftouhi, Where to place a spherical obstacle so as to maximize the first Steklov eigenvalue, hal: 02334941. doi: 10.13140/RG.2.2.12780.72324.

[9]

B. Fuglede, Stability in the isoperimetric problem for convex or nearly spherical domains in $ \mathbb{R}^n$, Trans. Amer. Math. Soc., 314 (1989), 619-638.  doi: 10.2307/2001401.

[10]

J. Hersch, Contribution to The Method of Interior Parallels Applied to Vibrating Membranes, Stanford Univ. Press, Stanford, Calif. (1962), 132-139.

[11]

G. Paoli, G. Piscitelli and L. Trani, Sharp estimates for the first $p$-laplacian eigenvalue and for the $p$-torsional rigidity on convex sets with holes, ESAIM. Contr. Optim Calc. Var., (in press). doi: 10.1051/cocv/2020033.

[12]

L. E. Payne and H. F. Weinberger, Some isoperimetric inequalities for membrane frequencies and torsional rigidity, J. Math. Anal. Appl., 2 (1961), 210-216.  doi: 10.1016/0022-247X(61)90031-2.

[13]

G. Santhanam and S. Verma, On Eigenvalue Problems Related to the Laplacian in a Class of Doubly Connected Domains, arXiv: 1803.05750.

[14]

R. Weinstock, Inequalities for a classical eigenvalue problem, J. Rational Mech. Anal., 3 (1954), 745-753.  doi: 10.1512/iumj.1954.3.53036.

show all references

References:
[1]

F. Brock, An isoperimetric inequality for eigenvalues of the Stekloff problem, Z. Angew. Math. Mech., 81 (2001), 69-71. 

[2]

D. Bucur, V. Ferone, C. Nitsch, C. Trombetti, Weinstock inequality in higher dimensions, J. Differential Geom. (in press). arXiv: 1710.04587

[3]

J. ChoeM. Ghomi and M. Ritoré, The relative isoperimetric inequality outside convex domains in $\mathbb{R}^n$, Calc. Var. Partial Differ. Equ., 29 (2007), 421-429.  doi: 10.1007/s00526-006-0027-z.

[4]

G. CrastaI. Fragalá and F. Gazzola, A sharp upper bound for the torsional rigidity of rods by means of web functions, Arch. Ration. Mech. Anal., 164 (2002), 189-211.  doi: 10.1007/s002050200205.

[5]

F. Della Pietra and G. Piscitelli, An optimal bound for nonlinear eigenvalues and torsional rigidity on domains with holes, Milan J. Math., (2020), 12pp. doi: 10.1007/s00032-020-00320-9.

[6]

M. Egert and P. Tolksdorf, Characterizations of Sobolev functions that vanish on a part of the boundary, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 729-743.  doi: 10.3934/dcdss.2017037.

[7]

V. FeroneC. Nitsch and C. Trombetti, On a conjectured reverse Faber-Krahn inequality for a Steklov-type Laplacian eigenvalue, Commun. Pure Appl. Anal., 14 (2015), 63-82.  doi: 10.3934/cpaa.2015.14.63.

[8]

I. Ftouhi, Where to place a spherical obstacle so as to maximize the first Steklov eigenvalue, hal: 02334941. doi: 10.13140/RG.2.2.12780.72324.

[9]

B. Fuglede, Stability in the isoperimetric problem for convex or nearly spherical domains in $ \mathbb{R}^n$, Trans. Amer. Math. Soc., 314 (1989), 619-638.  doi: 10.2307/2001401.

[10]

J. Hersch, Contribution to The Method of Interior Parallels Applied to Vibrating Membranes, Stanford Univ. Press, Stanford, Calif. (1962), 132-139.

[11]

G. Paoli, G. Piscitelli and L. Trani, Sharp estimates for the first $p$-laplacian eigenvalue and for the $p$-torsional rigidity on convex sets with holes, ESAIM. Contr. Optim Calc. Var., (in press). doi: 10.1051/cocv/2020033.

[12]

L. E. Payne and H. F. Weinberger, Some isoperimetric inequalities for membrane frequencies and torsional rigidity, J. Math. Anal. Appl., 2 (1961), 210-216.  doi: 10.1016/0022-247X(61)90031-2.

[13]

G. Santhanam and S. Verma, On Eigenvalue Problems Related to the Laplacian in a Class of Doubly Connected Domains, arXiv: 1803.05750.

[14]

R. Weinstock, Inequalities for a classical eigenvalue problem, J. Rational Mech. Anal., 3 (1954), 745-753.  doi: 10.1512/iumj.1954.3.53036.

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