• Previous Article
    Bubbling solutions for the Liouville equation around a quantized singularity in symmetric domains
  • CPAA Home
  • This Issue
  • Next Article
    Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential
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  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 & 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.   Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

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

show all references

References:
[1]

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

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

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

[1]

Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533

[2]

Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209

[3]

Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137

[4]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825

[5]

Jon Aaronson, Dalia Terhesiu. Local limit theorems for suspended semiflows. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6575-6609. doi: 10.3934/dcds.2020294

[6]

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

[7]

Demetres D. Kouvatsos, Jumma S. Alanazi, Kevin Smith. A unified ME algorithm for arbitrary open QNMs with mixed blocking mechanisms. Numerical Algebra, Control & Optimization, 2011, 1 (4) : 781-816. doi: 10.3934/naco.2011.1.781

[8]

Chaoqian Li, Yajun Liu, Yaotang Li. Note on $ Z $-eigenvalue inclusion theorems for tensors. Journal of Industrial & Management Optimization, 2021, 17 (2) : 687-693. doi: 10.3934/jimo.2019129

[9]

Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397

[10]

Seung-Yeal Ha, Shi Jin. Local sensitivity analysis for the Cucker-Smale model with random inputs. Kinetic & Related Models, 2018, 11 (4) : 859-889. doi: 10.3934/krm.2018034

[11]

Valery Y. Glizer. Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020027

[12]

Samir Adly, Oanh Chau, Mohamed Rochdi. Solvability of a class of thermal dynamical contact problems with subdifferential conditions. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 91-104. doi: 10.3934/naco.2012.2.91

[13]

M. Mahalingam, Parag Ravindran, U. Saravanan, K. R. Rajagopal. Two boundary value problems involving an inhomogeneous viscoelastic solid. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1351-1373. doi: 10.3934/dcdss.2017072

[14]

Feng Luo. A combinatorial curvature flow for compact 3-manifolds with boundary. Electronic Research Announcements, 2005, 11: 12-20.

[15]

Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223

[16]

Mats Gyllenberg, Jifa Jiang, Lei Niu, Ping Yan. On the classification of generalized competitive Atkinson-Allen models via the dynamics on the boundary of the carrying simplex. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 615-650. doi: 10.3934/dcds.2018027

[17]

Teddy Pichard. A moment closure based on a projection on the boundary of the realizability domain: 1D case. Kinetic & Related Models, 2020, 13 (6) : 1243-1280. doi: 10.3934/krm.2020045

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (101)
  • HTML views (73)
  • Cited by (0)

[Back to Top]