Spectral partitions for Sturm–Liouville problems

Abstract

We look for best partitions of the unit interval that minimize certain functionals defined in terms of the eigenvalues of Sturm-Liouville problems. Via Γ-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.

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).…”

“…Notice that the extension to the sub-homogeneous case p < q is not immediate anyhow: one has to face with the locality of the functional C(Ω, Σ) (see Proposition 3.2), which requires several ad hoc arguments. Now, as the space V 1 (Ω) is compact in the weak-* topology, from Γ-convergence theory (see [14] and also [28,Section 5]) we can recover some information on the asymptotic behaviour of the minimizers of (4). Corollary 2.3.…”

“…are satisfied as soon as k is large enough (depending only on ε). The finiteness of the limit in (28) implies that C(Ω, Σ L ) → 0, and this in turn forces Σ L to converge to Ω in the Hausdorff metric (otherwise, a subsequence among the open sets Ω \ Σ L would contain a ball B r (x L ) of fixed radius r > 0, and by Remark 3.1 we would have C(Ω, Σ L ) ≥ C(Ω, Σ L ∪ ∂B r (x L )) ≥ C(B r (x L ), ∂B r (x L )): by (1) this bound would be uniform in L, a contradiction). Since Σ L is a closed connected set, this also entails that (see, e.g., [22])…”

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

“…where {x n j } is the family of points that identify the optimal partition {I n j } and δ x denotes the Dirac delta supported at a point x ∈ I (the normalization factor n − 1 guarantees µ n to be a probability measure). Notice that this association is a common strategy that has been used in similar contexts (see for instance [5,7,28,29,30]). Now if n → ∞, by the compactness of the space of probability measures P(I), we may assume that, up to subsequences, the probability measures {µ n } defined in (8) weakly* converge to some probability measure µ.…”

“…Then let f be the set function (λ r,t 1 ) 1/r . For Definition 3.1 we can refer to the limit proved in [29,Lemma 2.1] where it has been identified the function s = w/p. Definition 3.2 holds thanks to the boundedness assumptions on I and on the coefficients p, q, w. Indeed, by (17) we have…”