On a three-dimensional model for MEMS with hinged boundary conditions

We study a free boundary problem arising from the modeling of an idealized electrostatically actuated MEMS device. In contrast to existing literature, we consider a three-dimensional device involving a hinged elastic plate. The model couples the electrostatic potential to the displacement of the elastic plate, which is caused by a voltage difference that is applied to the device. The electrostatic potential is harmonic in the free domain between the elastic plate and a rigid ground plate. The elastic plate displacement solves a fourth-order parabolic equation with hinged boundary conditions and a right-hand side proportional to the gradient trace of the electrostatic potential on the elastic plate. We establish local and global well-posedness of the model in dependence of the applied voltage difference and show that only touchdown of the elastic plate on the ground plate can generate a finite time singularity. Next, we consider stationary solutions and prove that such solutions exist for small voltage values and do not exist for large voltage values. To prove the nonexistence result, we show that the fourth-order elliptic operator with hinged boundary conditions satisfies a positivity preserving property and has a positive eigenpair

Symmetry and rigidity for the hinged composite plate problem

The composite plate problem is an eigenvalue optimization problem related to the fourth order operator $(-\Delta)^2$. In this paper we continue the study started in [10], focusing on symmetry and rigidity issues in the case of the hinged composite plate problem, a specific situation that allows us to exploit classical techniques like the moving plane method.Comment: 19 pages, 2 figure

Symmetry in the composite plate problem

In this paper we deal with the composite plate problem, namely the following optimization eigenvalue problem $$\inf_{\rho \in \mathrm{P}} \inf_{u \in \mathcal{W}\setminus\{0\}} \frac{\int_{\Omega}(\Delta u)^2}{\int_{\Omega} \rho u^2},$$ where $\mathrm{P}$ is a class of admissible densities, $\mathcal{W}= H^{2}_{0}(\Omega)$ for Dirichlet boundary conditions and $\mathcal W= H^2(\Omega) \cap H^1_{0}(\Omega)$ for Navier boundary conditions. The associated Euler-Lagrange equation is a fourth-order elliptic PDE governed by the biharmonic operator $\Delta^2$. In the spirit of [10], we study qualitative properties of the optimal pairs $(u,\rho)$. In particular, we prove existence and regularity and we find the explicit expression of $\rho$. When $\Omega$ is a ball, we can also prove uniqueness of the optimal pair, as well as positivity of $u$ and radial symmetry of both $u$ and $\rho$.Comment: 26 page

Symmetry and rigidity results for composite membranes and plates

The composite membrane problem is an eigenvalue optimization problem deeply studied from the beginning of the '00's. In this note we survey most of the results proved by several authors over the last twenty years, up to the recent paper [14] written in collaboration with Giovanni Cupini.We finally introduce an eigenvalue optimization problem for a fourth order operator, called composite plate problem and we present the symmetry and rigidity results obtained in this framework. These last mentioned results are part of the papers [12,13], written in collaboration with Francesca Colasuonno.Il problema della membrana composita è un problema di ottimizzazione di autovalori i cui primi contributi risalgono agli inizi degli anni '00. In questa nota presentiamo una sintesi dei principali risultati ottenuti negli ultimi venti anni, fino al recente contributo [14] scritto in collaborazione con Giovanni Cupini.Introdurremo poi un problema di ottimizzazione di autovalori per un operatore del quart'ordine noto come problema della piastra composita, e presenteremo alcuni risultati di simmetria e rigidità in questo ambito. Questi ultimi risultati sono contenuti nei lavori [12,13] scritti in collaborazione con Francesca Colasuonno

A hinged plate equation and iterated Dirichlet Laplace operator on domains with concave corners

AbstractFourth order hinged plate type problems are usually solved via a system of two second order equations. For smooth domains such an approach can be justified. However, when the domain has a concave corner the bi-Laplace problem with Navier boundary conditions may have two different types of solutions, namely u1 with u1,Δu1∈H˚1 and u2∈H2∩H˚1. We will compare these two solutions. A striking difference is that in general only the first solution, obtained by decoupling into a system, preserves positivity, that is, a positive source implies that the solution is positive. The other type of solution is more relevant in the context of the hinged plate. We will also address the higher-dimensional case. Our main analytical tools will be the weighted Sobolev spaces that originate from Kondratiev. In two dimensions we will show an alternative that uses conformal transformation. Next to rigorous proofs the results are illustrated by some numerical experiments for planar domains

Comparison and sign preserving properties of bilaplace boundary value problems in domains with corners

This work is focused on the study of the Kirchhoff-Love model for thin, transversally loaded plates with corner singularities on the boundary. The former consists in finding a real valued function u, defined in a bounded, planar set. The latter represents the shape of the plate and u(x) its vertical deflection at the point x. This makes sense since we are in the framework of linear elasticity, that is, the model assumes that no horizontal deformation takes place. The function u is found as the minimizer of the Kirchhoff energy functional in different subsets of a suitable Sobolev space, incorporating the boundary conditions. One can distinguish the following cases: (i) clamped: the function u and its exterior normal derivative are assumed to be zero on the boundary of the plate, (ii) hinged: we assume only u=0 on the boundary and (iii) supported: u is assumed nonnegative on the boundary. A hinged plate will additionally satisfy a set of natural boundary conditions, whereas a solution in the supported case will exist only if we assume that the load f pushes the plate down effectively; in that case a set of natural boundary conditions will be again fulfilled. It is however common within the mathematical and engineering literature to confuse the hinged and supported plates. This originates from the expectation that when pressed down, a supported plate, like a supported beam, will have a zero deflection on the boundary. Here we prove the contrary: If the domain has a corner, then a hinged plate cannot be in general a minimizer of the energy functional if we allow variations with positive boundary values. Moreover, we illustrate that a hinged plate with a sufficiently smooth boundary satisfies a comparison principle. In the last chapter we consider the problem of decoupling a clamped plate into a system of second order equations. This approach is very important for numerical procedures, since one can then use standard piecewise linear elements. We show that such a decomposition yields the correct solutions only if the domain has convex corners; when a concave corner is present then the system has no solution