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

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

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

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

Asymptotics of the First Laplace Eigenvalue with Dirichlet Regions of Prescribed Length

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 ΣL 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