6 papers
Neumann's nodal line may be closed on doubly-connected planar domains
Pedro Freitas, Roméo Leylekian
We show the existence of planar domains with one hole for which the first non-trivial Neumann eigenfunction has a closed nodal line fully contained inside the domain. This is optim…
Payne's nodal line conjecture fails on doubly-connected planar domains
Pedro Freitas, Roméo Leylekian
We present examples of bounded planar domains with one single hole for which the nodal line of a second Dirichlet eigenfunction is closed and does not touch the boundary. This show…
Extremal problems for clamped plates under tension
Pedro Freitas, Roméo Leylekian
We address extremum problems for spectral quantities associated with operators of the form with Dirichlet boundary conditions, for non-negative values of . The focu…
Fourth order Saint-Venant inequalities: maximizing compliance and mean deflection among clamped plates
Mark Ashbaugh, Dorin Bucur, Richard S. Laugesen +1
We prove a fourth order analogue of the Saint-Venant inequality: the mean deflection of a clamped plate under uniform transverse load is maximal for the ball, among plates of presc…
Existence of an optimal shape for the first eigenvalue of polyharmonic operators
Roméo Leylekian
We prove the existence of an open set minimizing the first eigenvalue of the Dirichlet polylaplacian of order under volume constraint. Moreover, the corresponding eigenfun…
Sufficient conditions yielding the Rayleigh Conjecture for the clamped plate
Roméo Leylekian
The Rayleigh Conjecture for the bilaplacian consists in showing that the clamped plate with least principal eigenvalue is the ball. The conjecture has been shown to hold in 1995 by…