Analyticity and Criticality Results for the Eigenvalues of the Biharmonic Operator

Abstract

Abstract:We consider the eigenvalues of the biharmonic operator subject to several homogeneous boundary conditions (Dirichlet, Neumann, Navier, Steklov). We show that simple eigenvalues and elementary symmetric functions of multiple eigenvalues are real analytic, and provide Hadamard-type formulas for the corresponding shape derivatives. After recalling the known results in shape optimization, we prove that balls are always critical domains under volume constraint.

A note on the Neumann eigenvalues of the biharmonic operator

“…We also refer to , where it is discussed the dependence of the eigenvalues of polyharmonic operators upon variation of the mass density, and to where the authors consider Neumann and Steklov‐type eigenvalue problems for the biharmonic operator with particular attention to shape optimization and mass concentration phenomena. We also mention , where the author considers the shape sensitivity problem for the eigenvalues of the biharmonic operator (in particular, also those of problem ) for . We note that other issues have been addressed in the literature for polyharmonic operators, such as analyticity, continuity, and stability estimates for the eigenvalues with respect to the shape; we refer to and the references therein.…”

“…We also refer to [5], where it is discussed the dependence of the eigenvalues of polyharmonic operators upon variation of the mass density, and to [6] where the authors consider Neumann and Steklov-type eigenvalue problems for the biharmonic operator with particular attention to shape optimization and mass concentration phenomena. We also mention [7], where the author considers the shape sensitivity problem for the eigenvalues of the biharmonic operator (in particular, also those of problem (1)) for 2…”

Extremal eigenvalues of the Dirichlet biharmonic operator on rectangles

“…Then the sequence of optimal domains Ω * k converges to the regular hexagon with perimeter α. The proof of Theorem 5.1 goes along the same lines of the correponding results in [10], now using the first term in the Weyl asymptotics (2), and inequalities (4) and (11). We also note that this result can be extended to a general polyharmonic problem of the form (14) (−∆) m u = λu, in Ω, u = ∂u ∂ν = · · · = ∂ m−1 u ∂ν m−1 = 0, on ∂Ω, for m ≥ 1, as formulas (2) and (4) can be generalized to this case as well.…”

“…1). We also note that, even though the general form of the shape derivative for eigenvalues of problem (1) is known (see e.g., [11,12]), its value is extremely difficult to estimate for the square and for rectangles in general, since, in contrast with the Dirichlet Laplacian and as mentioned above, the explicit form of the eigenfunctions is not known. Figure 1.…”

On the Behavior of Clamped Plates under Large Compression

“…Interestingly enough, though, eigenfunctions on the ball always satisfy condition (5.7). For a more detailed analysis of this fact, we refer to [9,10]. 6.…”