We consider the problem of maximizing the first eigenvalue of the p-Laplacian (possibly with nonconstant coefficients) over a fixed domain Ω, with Dirichlet conditions along ∂Ω and along a supplementary set Σ, which is the unknown of the optimization problem. The set Σ, which plays the role of a supplementary stiffening rib for a membrane Ω, is a compact connected set (e.g., a curve or a connected system of curves) that can be placed anywhere in Ω and is subject to the constraint of an upper bound L to its total length (one-dimensional Hausdorff measure). This upper bound prevents Σ from spreading throughout Ω and makes the problem well-posed. We investigate the behavior of optimal sets Σ L as L → ∞ via Γ-convergence, and we explicitly construct certain asymptotically optimal configurations. We also study the behavior as p → ∞ with L fixed, finding connections with maximum-distance problems related to the principal frequency of the ∞-Laplacian.

We use a gap function in order to compare the torsional performances of different reinforced plates under the action of external forces. Then, we address a shape optimization problem, whose target is to minimize the torsional displacements of the plate: this leads us to set up a minimaxmax problem, which includes a new kind of worst-case optimization. Two kinds of reinforcements are considered: one aims at strengthening the plate, the other aims at weakening the action of the external forces. For both of them, we study the existence of optima within suitable classes of external forces and reinforcements. Our results are complemented with numerical experiments and with a number of open problems and conjectures.2010 Mathematics Subject Classification. 35J40; 35P15; 74K20.

Abstract. We study a shape optimization problem for the first eigenvalue of an elliptic operator in divergence form, with non constant coefficients, over a fixed domain Ω. Dirichlet conditions are imposed along ∂Ω and, in addition, along a set Σ of prescribed length (1-dimensional Hausdorff measure). We look for the best shape and position for the supplementary Dirichlet region Σ in order to maximize the first eigenvalue. The limit distribution of the optimal sets, as their prescribed length tends to infinity, is characterized via Γ-convergence of suitable functionals defined over varifolds: the use of varifolds, as opposed to probability measures, allows one to keep track of the local orientation of the optimal sets (which comply with the anisotropy of the problem), and not just of their limit distribution.

Inside a fixed bounded domain Ω of the plane, we look for the best compact connected set K, of given perimeter, in order to maximize the first Dirichlet eigenvalue λ 1 (Ω \ K). We discuss some of the qualitative properties of the maximizers, moving toward existence, regularity and geometry. Then we study the problem in specific domains: disks, rings, and, more generally, disks with convex holes. In these situations, we prove symmetry and in some cases non symmetry results, identifying the solution.We choose to work with the outer Minkowski content as the "good" notion of perimeter. Therefore, we are led to prove some new properties for it as its lower semicontinuity with respect to the Hausdorff convergence and the fact that the outer Minkowski content is equal to the Hausdorff lower semicontinuous envelope of the classical perimeter.

The local solvability of the Cauchy problem for the nonlinear vibrating plate equation is showed in the framework of modulation spaces. In the opposite direction, it is proved that there is no local wellposedness in Wiener amalgam spaces even for the solution to the homogeneous vibrating plate equation.