Convex shape optimization for the least biharmonic Steklov eigenvalue

Abstract

Abstract. The least Steklov eigenvalue d1 for the biharmonic operator in bounded domains gives a bound for the positivity preserving property for the hinged plate problem, appears as a norm of a suitable trace operator, and gives the optimal constant to estimate the L 2 -norm of harmonic functions. These applications suggest to address the problem of minimizing d1 in suitable classes of domains. We survey the existing results and conjectures about this topic; in particular, the existence of a convex domain of fixed measure minimizing d1 is known, although the optimal shape is still unknown. We perform several numerical experiments which strongly suggest that the optimal planar shape is the regular pentagon. We prove the existence of a domain minimizing d1 also among convex domains having fixed perimeter and present some numerical results supporting the conjecture that, among planar domains, the disk is the minimizer.

Analyticity and Criticality Results for the Eigenvalues of the Biharmonic Operator

“…Nevertheless, in [7] the authors can prove that, among all convex domain of fixed measure there exists a minimizer, but nothing is known about the optimal shape, or if the convexity assumption can be relaxed. We also refer to [2,5,6] for other results on problem (5). Another Steklov problem for the biharmonic operator which has appeared very recently in [15] (see also [16]) is the following…”

On the variation of longitudinal and torsional frequencies in a partially hinged rectangular plate

“…Now we observe that all the eigenfunctions show symmetry properties, i.e., they are either even or odd in the y variable, and symmetric or skew-symmetric with respect to x, see Theorem 2.1. In particular this implies that |D 2 v| 2 , (∆v) 2 and v 2 are equal for y = ± and symmetric with respect to x = π/2. This proves formula (13) and completes the proof of Theorem 3.3.…”

“…The boundary conditions on the large edges model the fact that the deck is free to move vertically and they may also be written in the equivalent form (cf. [25]) (2) (1 − σ) ∂ 2 u ∂ν 2 + σ∆u = ∂∆u ∂ν + (1 − σ)div ∂Ω (ν · D 2 u) ∂Ω = 0 on (0, π) × {− , }.…”

“…These functionals do not simply depend on the eigenvalues but also on the shape of the domain: in these situations, the analysis usually requires a suitable combination of variational methods from shape optimization and numerical methods, see e.g. [2,34,42,50] for some examples. We show that some of these functionals partially fulfill the basic rules of a shape functional aiming to compute the energy threshold for the torsional stability.…”

Optimality conditions of the first eigenvalue of a fourth order Steklov problem

“…Let us summarize some properties which are well known by now. We refer to [1,3,4,5] and [6] for more details.…”