Computational Methods for Extremal Steklov Problems

We develop a computational method for extremal Steklov eigenvalue problems and apply it to study the problem of maximizing the $p$th Steklov eigenvalue as a function of the domain with a volume constraint. In contrast to the optimal domains for several other extremal Dirichlet- and Neumann-Laplacian eigenvalue problems, computational results suggest that the optimal domains for this problem are very structured. We reach the conjecture that the domain maximizing the $p$th Steklov eigenvalue is unique (up to dilations and rigid transformations), has $p$-fold symmetry, and has at least one axis of symmetry. The $p$th Steklov eigenvalue has multiplicity 2 if $p$ is even and multiplicity 3 if $p\geq3$ is odd.

[1]  Robert K. Brayton,et al.  A new algorithm for statistical circuit design based on quasi-newton methods and function splitting , 1979 .

[2]  Edouard Oudet,et al.  Numerical minimization of eigenmodes of a membrane with respect to the domain , 2004 .

[3]  C. Kao,et al.  Minimal Convex Combinations of Three Sequential Laplace-Dirichlet Eigenvalues , 2013 .

[4]  Pan Cheng,et al.  Nyström methods and extrapolation for solving Steklov eigensolutions and its application in elasticity , 2012 .

[5]  Pedro Freitas,et al.  Optimisation of Eigenvalues of the Dirichlet Laplacian with a Surface Area Restriction , 2016 .

[6]  Friedemann Brock,et al.  An Isoperimetric Inequality for Eigenvalues of the Stekloff Problem , 2001 .

[7]  R. Banuelos,et al.  Eigenvalue inequalities for mixed Steklov problems , 2009 .

[8]  B. A. Troesch,et al.  An isoperimetric sloshing problem , 1965 .

[9]  Iosif Polterovich,et al.  The legacy of Vladimir Andreevich Steklov , 2014 .

[10]  Antoine Henrot,et al.  Extremum Problems for Eigenvalues of Elliptic Operators , 2006 .

[11]  Iosif Polterovich,et al.  Shape optimization for low Neumann and Steklov eigenvalues , 2008, 0811.2617.

[12]  Ahmad El Soufi,et al.  Isoperimetric control of the Steklov spectrum , 2011, 1103.2863.

[13]  Braxton Osting,et al.  Optimization of spectral functions of Dirichlet-Laplacian eigenvalues , 2010, J. Comput. Phys..

[14]  Bodo Dittmar,et al.  Sums of reciprocal Stekloff eigenvalues , 2004 .

[15]  R. Kress,et al.  Inverse Acoustic and Electromagnetic Scattering Theory , 1992 .

[16]  R. Laugesen,et al.  Steklov Eigenvalues and Quasiconformal Maps of Simply Connected Planar Domains , 2014, 1412.8073.

[17]  Édouard Oudet,et al.  Qualitative and Numerical Analysis of a Spectral Problem with Perimeter Constraint , 2016, SIAM J. Control. Optim..

[18]  Richard Schoen,et al.  The first Steklov eigenvalue, conformal geometry, and minimal surfaces , 2009, 0912.5392.

[19]  R. Kress.  Linear Integral Equations , 1989 .

[20]  Robert Weinstock,et al.  Inequalities for a Classical Eigenvalue Problem , 1954 .

[21]  Eldar Akhmetgaliyev,et al.  Fast Numerical Methods for Mixed, Singular Helmholtz Boundary Value Problems and Laplace Eigenvalue Problems - with Applications to Antenna Design, Sloshing, Electromagnetic Scattering and Spectral Geometry , 2016 .

[22]  Menahem Schiffer,et al.  Some inequalities for Stekloff eigenvalues , 1974 .

[23]  Pedro Freitas,et al.  Numerical Optimization of Low Eigenvalues of the Dirichlet and Neumann Laplacians , 2012, J. Optim. Theory Appl..

[24]  Beniamin Bogosel,et al.  The method of fundamental solutions applied to boundary eigenvalue problems , 2016, J. Comput. Appl. Math..

[25]  JinHuang,et al.  THE MECHANICAL QUADRATURE METHODS AND THEIR EXTRAPOLATION FOR SOLVING BIE OF STEKLOV EIGENVALUE PROBLEMS , 2004 .

[26]  Pedro R. S. Antunes.  Optimization of sums and quotients of Dirichlet-Laplacian eigenvalues , 2013, Appl. Math. Comput..

[27]  R. Ibrahim.  Liquid Sloshing Dynamics: Theory and Applications , 2005 .

[28]  Tadeusz Kulczycki,et al.  The shape of the fundamental sloshing mode in axisymmetric containers , 2014 .

[29]  Julián Fernández Bonder,et al.  Optimization of the first Steklov eigenvalue in domains with holes: a shape derivative approach , 2006, Annali di Matematica Pura ed Applicata.

[30]  Iosif Polterovich,et al.  SPECTRAL GEOMETRY OF THE STEKLOV PROBLEM , 2014, 1411.6567.

[31]  Jimmy Lamboley,et al.  An extremal eigenvalue problem for the Wentzell-Laplace operator , 2014, 1401.7098.

[32]  Rongjie Lai,et al.  Maximization of Laplace−Beltrami eigenvalues on closed Riemannian surfaces , 2014, 1405.4944.