# The Bilaplacian with Robin boundary conditions

@inproceedings{Buoso2021TheBW, title={The Bilaplacian with Robin boundary conditions}, author={Davide Buoso and James B. Kennedy}, year={2021} }

We introduce Robin boundary conditions for biharmonic operators, which are a model for elastically supported plates and are closely related to the study of spaces of traces of Sobolev functions. We study the dependence of the operator, its eigenvalues, and eigenfunctions on the Robin parameters. We show in particular that when the parameters go to plus infinity the Robin problem converges to other biharmonic problems, and obtain estimates on the rate of divergence when the parameters go to… Expand

#### 4 Citations

Semiclassical bounds for spectra of biharmonic operators.

- Mathematics
- 2019

The averaged variational principle (AVP) is applied to various biharmonic operators. For the Riesz mean $R_1(z)$ of the eigenvalues we improve the known sharp semiclassical bounds in terms of the… Expand

Positivity for the clamped plate equation under high tension

- Mathematics
- 2021

In this article we consider positivity issues for the clamped plate equation with high tension γ > 0. This equation is given by ∆2u − γ∆u = f under clamped boundary conditions. Here we show, that… Expand

Two inequalities for the first Robin eigenvalue of the Finsler Laplacian

- Mathematics
- 2021

Abstract. Let Ω Ă R, n ě 2, be a bounded connected, open set with Lipschitz boundary. Let F be a suitable norm in R and let ∆Fu “ div pFξp∇uqF p∇uqq be the so-colled Finsler Laplacian, with u P HpΩq.… Expand

A Sharp Isoperimetric Inequality for the Second Eigenvalue of the Robin Plate

- Mathematics
- 2020

Among all $C^{\infty}$ bounded domains with equal volume, we show that the second eigenvalue of the Robin plate is uniquely maximized by an open ball, so long as the Robin parameter lies within a… Expand

#### References

SHOWING 1-10 OF 78 REFERENCES

Spectral Analysis of the Biharmonic Operator Subject to Neumann Boundary Conditions on Dumbbell Domains

- Mathematics
- 2017

We consider the biharmonic operator subject to homogeneous boundary conditions of Neumann type on a planar dumbbell domain which consists of two disjoint domains connected by a thin channel. We… Expand

Analyticity and Criticality Results for the Eigenvalues of the Biharmonic Operator

- Mathematics
- 2016

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… Expand

Higher order elliptic operators on variable domains. Stability results and boundary oscillations for intermediate problems

- Mathematics
- 2017

Abstract We study the spectral behavior of higher order elliptic operators upon domain perturbation. We prove general spectral stability results for Dirichlet, Neumann and intermediate boundary… Expand

A note on the Neumann eigenvalues of the biharmonic operator

- Mathematics
- 2016

We study the dependence of the eigenvalues of the biharmonic operator subject to Neumann boundary conditions on the Poisson's ratio. In particular, we prove that the Neumann eigenvalues are Lipschitz… Expand

On the eigenvalues of a Robin problem with a large parameter

- Mathematics
- 2014

We consider the Robin eigenvalue problem ∆u+λu = 0 in Ω, ∂u/∂ν +αu = 0 on ∂Ω where Ω ⊂ R, n > 2 is a bounded domain and α is a real parameter. We investigate the behavior of the eigenvalues λk(α) of… Expand

On a classical spectral optimization problem in linear elasticity

- Mathematics
- 2014

We consider a classical shape optimization problem for the eigenvalues of elliptic operators with homogeneous boundary conditions on domains in the N-dimensional Euclidean space. We survey recent… Expand

Steklov-type eigenvalues associated with best Sobolev trace constants: domain perturbation and overdetermined systems

- Mathematics
- 2014

We consider a variant of the classic Steklov eigenvalue problem, which arises in the study of the best trace constant for functions in Sobolev space. We prove that the elementary symmetric functions… Expand

On the estimates of eigenvalues of the boundary value problem with large parameter

- Mathematics
- 2015

Abstract We consider the eigenvalue problem Δu + λu = 0 in Ω with Robin condition + αu = 0 on ∂Ω , where Ω ⊂ Rn , n ≥ 2 is a bounded domain and α is a real parameter. We obtain the estimates to the… Expand

Spectral stability for a class of fourth order Steklov problems under domain perturbations

- Mathematics
- Calculus of Variations and Partial Differential Equations
- 2019

We study the spectral stability of two fourth order Steklov problems upon domain perturbation. One of the two problems is the classical DBS—Dirichlet Biharmonic Steklov—problem, the other one is a… Expand

Bounds and extremal domains for Robin eigenvalues with negative boundary parameter

- Mathematics, Physics
- 2016

Abstract We present some new bounds for the first Robin eigenvalue with a negative boundary parameter. These include the constant volume problem, where the bounds are based on the shrinking… Expand