# American Institute of Mathematical Sciences

September  2008, 7(5): 1193-1201. doi: 10.3934/cpaa.2008.7.1193

## Where to place a spherical obstacle so as to maximize the second Dirichlet eigenvalue

 1 Laboratoire de Mathématiques et Physique Théorique, UMR CNRS 6083, Université François Rabelais de Tours, Parc de Grandmont, F-37200 Tours, France, France

Received  November 2006 Revised  February 2008 Published  June 2008

We prove that among all doubly connected domains of $\mathbb R^n$ bounded by two spheres of given radii, the second eigenvalue of the Dirichlet Laplacian achieves its maximum when the spheres are concentric (spherical shell). The corresponding result for the first eigenvalue has been established by Hersch [12] in dimension 2, and by Harrell, Kröger and Kurata [10] and Kesavan [13] in any dimension.
We also prove that the same result remains valid when the ambient space $\mathbb R^n$ is replaced by the standard sphere $\mathbb S^n$ or the hyperbolic space $\mathbb H^n$.
Citation: Rola Kiwan, Ahmad El Soufi. Where to place a spherical obstacle so as to maximize the second Dirichlet eigenvalue. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1193-1201. doi: 10.3934/cpaa.2008.7.1193
 [1] 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 [2] Selma Yildirim Yolcu, Türkay Yolcu. Sharper estimates on the eigenvalues of Dirichlet fractional Laplacian. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2209-2225. doi: 10.3934/dcds.2015.35.2209 [3] Hua Chen, Hong-Ge Chen. Estimates the upper bounds of Dirichlet eigenvalues for fractional Laplacian. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 301-317. doi: 10.3934/dcds.2021117 [4] Fei-Ying Yang, Wan-Tong Li, Jian-Wen Sun. Principal eigenvalues for some nonlocal eigenvalue problems and applications. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 4027-4049. doi: 10.3934/dcds.2016.36.4027 [5] Mihai Mihăilescu. An eigenvalue problem possessing a continuous family of eigenvalues plus an isolated eigenvalue. Communications on Pure and Applied Analysis, 2011, 10 (2) : 701-708. doi: 10.3934/cpaa.2011.10.701 [6] Paolo Piersanti. On the improved interior regularity of a boundary value problem modelling the displacement of a linearly elastic elliptic membrane shell subject to an obstacle. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 1011-1037. doi: 10.3934/dcds.2021145 [7] Leandro M. Del Pezzo, Julio D. Rossi. Eigenvalues for a nonlocal pseudo $p-$Laplacian. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6737-6765. doi: 10.3934/dcds.2016093 [8] Lorenzo Brasco, Enea Parini, Marco Squassina. Stability of variational eigenvalues for the fractional $p-$Laplacian. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1813-1845. doi: 10.3934/dcds.2016.36.1813 [9] Craig Cowan, Pierpaolo Esposito, Nassif Ghoussoub. Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 1033-1050. doi: 10.3934/dcds.2010.28.1033 [10] Erwann Delay, Pieralberto Sicbaldi. Extremal domains for the first eigenvalue in a general compact Riemannian manifold. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5799-5825. doi: 10.3934/dcds.2015.35.5799 [11] Yansheng Zhong, Yongqing Li. On a p-Laplacian eigenvalue problem with supercritical exponent. Communications on Pure and Applied Analysis, 2019, 18 (1) : 227-236. doi: 10.3934/cpaa.2019012 [12] Giuseppina Barletta, Roberto Livrea, Nikolaos S. Papageorgiou. A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1075-1086. doi: 10.3934/cpaa.2014.13.1075 [13] Vladimir Georgiev, Koichi Taniguchi. On fractional Leibniz rule for Dirichlet Laplacian in exterior domain. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 1101-1115. doi: 10.3934/dcds.2019046 [14] Monique Dauge, Thomas Ourmières-Bonafos, Nicolas Raymond. Spectral asymptotics of the Dirichlet Laplacian in a conical layer. Communications on Pure and Applied Analysis, 2015, 14 (3) : 1239-1258. doi: 10.3934/cpaa.2015.14.1239 [15] Huicong Li, Jingyu Li. Asymptotic behavior of Dirichlet eigenvalues on a body coated by functionally graded material. Communications on Pure and Applied Analysis, 2017, 16 (4) : 1493-1516. doi: 10.3934/cpaa.2017071 [16] Changliang Zhou, Chunqin Zhou. Extremal functions of Moser-Trudinger inequality involving Finsler-Laplacian. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2309-2328. doi: 10.3934/cpaa.2018110 [17] Xavier Cabré, Manel Sanchón. Semi-stable and extremal solutions of reaction equations involving the $p$-Laplacian. Communications on Pure and Applied Analysis, 2007, 6 (1) : 43-67. doi: 10.3934/cpaa.2007.6.43 [18] Françoise Demengel, Thomas Dumas. Extremal functions for an embedding from some anisotropic space, involving the "one Laplacian". Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 1135-1155. doi: 10.3934/dcds.2019048 [19] Wolfgang Arendt, Rafe Mazzeo. Friedlander's eigenvalue inequalities and the Dirichlet-to-Neumann semigroup. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2201-2212. doi: 10.3934/cpaa.2012.11.2201 [20] Feng Du, Adriano Cavalcante Bezerra. Estimates for eigenvalues of a system of elliptic equations with drift and of bi-drifting laplacian. Communications on Pure and Applied Analysis, 2017, 6 (2) : 475-491. doi: 10.3934/cpaa.2017024

2021 Impact Factor: 1.273