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

Abstract

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.

Dirichlet conditions in Poincaré–Sobolev inequalities: the sub-homogeneous case

“…In the homogeneous case p = q the explicit formula given in Remark 3.4 for Poincaré-Sobolev constants with mixed boundary conditions is still true; this implies the validity of Theorem 3.6. Consequently, with only minor changes from the linear case p = q = 2, previously treated in [27,29] (see also [17,28] for other related results), one can prove the following result. Theorem 6.1 (Γ-convergence in the homogeneous case).…”

“…Let Σ ⊂ Ω be a compact set with N connected component for some N ∈ N. If the coefficient matrix A is constant and f = 1 then where BΩ and BΣ are the images of Ω and Σ through B. The above ratio is the same as the one considered in [27], but with numerator and denominator's powers decoupled. We may follow the same trick and test with the function…”

“…By (36), using (27) with ϕ(x, y) = η j (x)|A j y · y| 1 2 for all x ∈ Ω, y ∈ S 1 and some cutoff function η j ∈ C(Ω) such that 0 ≤ η j ≤ 1 and η j ≡ 1 over E j , we infer that lim sup…”

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

“…463-464]. Two dimensional versions of this maximization problem (19) have also been investigate in some recent works, see [18,28,30]. A possible physical interpretations of problem (19) is as follows.…”

“…The present paper starts from the easy consideration that most of the results in the literature are set up for optimal partition problems in a general higherdimensional framework. Our aim is, on the contrary, to focus on optimal partition problems in one dimension (in fact this is an ongoing project that we initiated in [28] looking for spectral partitions that minimize the sum of the eigenvalues of certain Sturm-Liouville problems). A crucial fact in one dimension is that each partition of an interval may be identified by the points that induce the partition.…”