Analyticity and criticality results for the eigenvalues of the biharmonic operator

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.Comment: To appear on the proceedings of the conference "Geometric Properties for Parabolic and Elliptic PDE's - 4th Italian-Japanese Workshop" held in Palinuro (Italy), May 25-29, 201

Optimal Constants in the Theory of Sobolev Spaces and PDEs

Recent research activities on sharp constants and optimal inequalities have shown their impact on a deeper understanding of geometric, analytical and other phenomena in the context of partial differential equations and mathematical physics. These intrinsic questions have applications not only to a-priori estimates or spectral theory but also to numerics, economics, optimization, etc

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

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