Extremal eigenvalues of the Dirichlet biharmonic operator on rectangles

Abstract

We study the behaviour of extremal eigenvalues of the Dirichlet biharmonic operator over rectangles with a given fixed area. We begin by proving that the principal eigenvalue is minimal for a rectangle for which the ratio between the longest and the shortest side lengths does not exceed 1.066459. We then consider the sequence formed by the minimal k th eigenvalue and show that the corresponding sequence of minimising rectangles converges to the square as k goes to infinity.2010 Mathematics Subject Classification. Primary 35J30. Secondary 35P15, 49R50, 74K20.

Two-term spectral asymptotics for the Dirichlet Laplacian in a Lipschitz domain

“…Therefore the second term on the right side of ( 18) is acceptable. Moreover, using the bound (17) and again Lemma A.3, we find…”

“…The interested reader is instead referred to [47] and references therein. Problems concerning the asymptotic behaviour of solutions of spectral shape optimization problems saw a rise in interest in recent years, largely motivated by a connection to the famous Pólya conjecture highlighted in [20] (see also [36]); see, for instance, [3,16,9,10,11,41,35,65,60,17] where a variety of problems of this type are studied. Some elements of our proofs.…”

Spectral optimization of inhomogeneous plates

“…The study of eigenmodes optimisation is central to the theory of inhomogeneous elastic plates and is of great applicative relevance. A vast literature has been devoted to the analysis of spectral optimisation problems for biharmonic operators, modelling plates of varying density and thickness under different settings [3,4,6,8,9,10,11,14,18,19]. In addition, several contributions are devoted to inverse problems arising in the study of such inhomogeneous plates [17,25,26].…”

The buckling eigenvalue problem in the annulus

“…While for second order problems these questions are trivial because of the maximum principle and other important tools such as the Krein-Rutman theory, their higher order counterparts have shown to be extremely difficult to approach and, apart from the case of corners [19], only very specific cases have been successfully studied, usually for the clamped plate problem (we refer the reader to [17] for an extensive discussion). Even estimates for the first eigenvalue on a rectangle turn out to be much more difficult to obtain than those for the Laplacian, see [6,22]. Regarding the buckling problem (1.3), even though the results in [19] still apply, very little is known for shapes different from balls, where the spectrum and the eigenspaces can be completely characterized in terms of Bessel functions.…”