A new proof of Friedman's second eigenvalue theorem and its extension to random lifts

2020 ◽  
Vol 53 (6) ◽  
pp. 1393-1439
Author(s):  
Charles BORDENAVE
Keyword(s):  
2010 ◽  
Vol 62 (4) ◽  
pp. 808-826
Author(s):  
Eveline Legendre

AbstractWe study extrema of the first and the second mixed eigenvalues of the Laplacian on the disk among some families of Dirichlet–Neumann boundary conditions. We show that the minimizer of the second eigenvalue among all mixed boundary conditions lies in a compact 1-parameter family for which an explicit description is given. Moreover, we prove that among all partitions of the boundary with bounded number of parts on which Dirichlet and Neumann conditions are imposed alternately, the first eigenvalue is maximized by the uniformly distributed partition.


2003 ◽  
Vol 131 (11) ◽  
pp. 3499-3505 ◽  
Author(s):  
Manuel del Pino ◽  
Jorge García-Melián ◽  
Monica Musso

2004 ◽  
Vol 06 (06) ◽  
pp. 901-912
Author(s):  
ANTONIO J. UREÑA

A celebrated result by Amann, Ambrosetti and Mancini [1] implies the connectedness of the region of existence for some parameter-depending boundary value problems which are resonant at the first eigenvalue. The analogous thing does not hold for problems which are resonant at the second eigenvalue.


2006 ◽  
Vol 416 (1) ◽  
pp. 175-183 ◽  
Author(s):  
Abraham Berman ◽  
Thomas Laffey ◽  
Arie Leizarowitz ◽  
Robert Shorten
Keyword(s):  

2005 ◽  
Vol 95 (2) ◽  
pp. 283-299 ◽  
Author(s):  
Yonatan Bilu ◽  
Nati Linial

1991 ◽  
Vol 58 (3) ◽  
pp. 759-765 ◽  
Author(s):  
Luis E. Suarez ◽  
Mahendra P. Singh

A mode synthesis approach is presented to calculate the eigenproperties of a structure from the eigenproperties of its substructures. The approach consists of synthesizing the substructures sequentially, one degree-of-freedom at a time. At each coupling stage, the eigenvalue is obtained as the solution of a characteristic equation, defined in closed form in terms of the eigenproperties obtained in the preceding coupling stage. The roots of the characteristic equation can be obtained by a simple Newton-Raphson root finding scheme. For each calculated eigenvalue, the eigenvector is defined by a simple closed-form expression. The eigenproperties obtained in the final coupling stage provide the desired eigenproperties of the coupled system. Thus, the approach avoids a conventional solution of the second eigenvalue problem. The approach can be implemented with the complete set or a truncated number of substructure modes; if the complete set of modes is used, the calculated eigenproperties would be exact. The approach can be used with any finite element discretization of structures. It requires only the free interface eigenproperties of the substructures. Successful application of the approach to a moderate size problem (255 degrees-of-freedom) on a microcomputer is also demonstrated.


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