scholarly journals Spectrum of the 1-Laplacian and Cheeger's Constant on Graphs

2015 ◽  
Vol 81 (2) ◽  
pp. 167-207 ◽  
Author(s):  
K. C. Chang
Keyword(s):  
Cheeger’S Constant

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).

scholarly journals Comparison estimates on the first eigenvalue of a quasilinear elliptic system

2020 ◽  
Vol 26 (2) ◽  
pp. 273-285
Author(s):  
Abimbola Abolarinwa ◽  
Shahroud Azami
Keyword(s):  
Riemannian Manifolds ◽  
Dirichlet Boundary ◽  
Eigenvalue Problems ◽  
Dirichlet Boundary Conditions ◽  
First Eigenvalue ◽  
Quasilinear Elliptic System ◽  
Type Estimate ◽  
The First Eigenvalue ◽  
Cheeger’S Constant ◽  
Quasilinear Elliptic

AbstractWe study a system of quasilinear eigenvalue problems with Dirichlet boundary conditions on complete compact Riemannian manifolds. In particular, Cheng comparison estimates and the inequality of Faber–Krahn for the first eigenvalue of a {(p,q)}-Laplacian are recovered. Lastly, we reprove a Cheeger-type estimate for the p-Laplacian, {1<p<\infty}, from where a lower bound estimate in terms of Cheeger’s constant for the first eigenvalue of a {(p,q)}-Laplacian is built. As a corollary, the first eigenvalue converges to Cheeger’s constant as {p,q\to 1,1}.

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