On the Existence of Metrics Which Maximize Laplace Eigenvalues on Surfaces

2017 ◽  
Vol 2018 (14) ◽  
pp. 4261-4355 ◽  
Author(s):  
Romain Petrides
Keyword(s):  
Laplace Eigenvalues

Laplace eigenvalues and bandwidth-type invariants of graphs

1993 ◽  
Vol 17 (3) ◽  
pp. 393-407 ◽  
Author(s):  
Martin Juvan ◽  
Bojan Mohar
Keyword(s):  
Laplace Eigenvalues

scholarly journals Guaranteed and Robust a Posteriori Bounds for Laplace Eigenvalues and Eigenvectors: Conforming Approximations

2017 ◽  
Vol 55 (5) ◽  
pp. 2228-2254 ◽  
Author(s):  
Eric Cancès ◽  
Geneviève Dusson ◽  
Yvon Maday ◽  
Benjamin Stamm ◽  
Martin Vohralík
Keyword(s):  
A Posteriori ◽  
Eigenvalues And Eigenvectors ◽  
Laplace Eigenvalues

scholarly journals Fundamental tone estimates for elliptic operators in divergence form and geometric applications

2006 ◽  
Vol 78 (3) ◽  
pp. 391-404 ◽  
Author(s):  
Gregório P. Bessa ◽  
Luquésio P. Jorge ◽  
Barnabé P. Lima ◽  
José F. Montenegro
Keyword(s):  
Lower Bounds ◽  
Vector Fields ◽  
Divergence Form ◽  
Elliptic Operators ◽  
Fundamental Tone ◽  
Space Forms ◽  
Closed Hypersurfaces ◽  
Laplace Eigenvalues ◽  
Cheeger’S Constant ◽  
Lr Operator

We establish a method for giving lower bounds for the fundamental tone of elliptic operators in divergence form in terms of the divergence of vector fields. We then apply this method to the Lr operator associated to immersed hypersurfaces with locally bounded (r + 1)-th mean curvature Hr + 1 of the space forms Nn+ 1(c) of constant sectional curvature c. As a corollary we give lower bounds for the extrinsic radius of closed hypersurfaces of Nn+ 1(c) with Hr + 1 > 0 in terms of the r-th and (r + 1)-th mean curvatures. Finally we observe that bounds for the Laplace eigenvalues essentially bound the eigenvalues of a self-adjoint elliptic differential operator in divergence form. This allows us to show that Cheeger's constant gives a lower bounds for the first nonzero Lr-eigenvalue of a closed hypersurface of Nn+ 1(c).

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