Dirichlet conditions in Poincar\'e-Sobolev inequalities: the sub-homogeneous case

We investigate the dependence of optimal constants in Poincar\'e- Sobolev inequalities of planar domains on the region where the Dirichlet condition is imposed. More precisely, we look for the best Dirichlet regions, among closed and connected sets with prescribed total length $L$ (one-dimensional Hausdorff measure), that make these constants as small as possible. We study their limiting behaviour, showing, in particular, that Dirichler regions homogenize inside the domain with comb-shaped structures, periodically distribuited at different scales and with different orientations. To keep track of these information we rely on a $\Gamma$-convergence result in the class of varifolds. This also permits applications to reinforcements of anisotropic elastic membranes. At last, we provide some evidences for a conjecture.Comment: arXiv admin note: text overlap with arXiv:1412.294

Optimal partitioning of an interval and applications to Sturm-Liouville eigenvalues

We study the optimal partitioning of a (possibly unbounded) interval of the real line into $n$ subintervals in order to minimize the maximum of certain set-functions, under rather general assumptions such as continuity, monotonicity, and a Radon-Nikodym property. We prove existence and uniqueness of a solution to this minimax partition problem, showing that the values of the set-functions on the intervals of any optimal partition must coincide. We also investigate the asymptotic distribution of the optimal partitions as $n$ tends to infinity. Several examples of set-functions fit in this framework, including measures, weighted distances and eigenvalues. We recover, in particular, some classical results of Sturm-Liouville theory: the asymptotic distribution of the zeros of the eigenfunctions, the asymptotics of the eigenvalues, and the celebrated Weyl law on the asymptotics of the counting function

The Cauchy Problem for the Vibrating Plate Equation in modulation spaces

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.Comment: 2 figures, some misprints correcte

Spectral partitions for Sturm-Liouville problems

We look for best partitions of the unit interval that minimize certain functionals defined in terms of the eigenvalues of Sturm-Liouville problems. Via \Gamma-convergence theory, we study the asymptotic distribution of the minimizers as the number of intervals of the partition tends to infinity. Then we discuss several examples that fit in our framework, such as the sum of (positive and negative) powers of the eigenvalues and an approximation of the trace of the heat Sturm-Liouville operator

A Minimaxmax Problem for Improving the Torsional Stability of Rectangular Plates

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

Where best to place a Dirichlet condition in an anisotropic membrane?

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 $\Omega$. Dirichlet conditions are imposed along $\partial \Omega$ and, in addition, along a set $\Sigma$ of prescribed length ($1$-dimensional Hausdorff measure). We look for the best shape and position for the supplementary Dirichlet region $\Sigma$ in order to maximize the first eigenvalue. The limit distribution of the optimal sets, as their prescribed length tends to infinity, is characterized via $\Gamma$-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.Comment: 23 pages, 2 figure

Transmission conditions obtained by homogenisation

Given a bounded open set in Rn, n 652, and a sequence (Kj) of compact sets converging to an (n-1)-dimensional manifold M, we study the asymptotic behaviour of the solutions to some minimum problems for integral functionals on \u3a9\Kj, with Neumann boundary conditions on 02(\u3a9\Kj). We prove that the limit of these solutions is a minimiser of the same functional on \u3a9\M subjected to a transmission condition on M, which can be expressed through a measure \ub5 supported on M. The class of all measures that can be obtained in this way is characterised, and the link between the measure \ub5 and the sequence (Kj) is expressed by means of suitable local minimum problems