Steiner symmetry in the minimization of the first eigenvalue of a fractional eigenvalue problem with indefinite weight

Abstract

Let Ω ⊂ R N , N ≥ 2, be an open bounded connected set. We consider the fractional weighted eigenvalue problem (−∆) s u = λρu in Ω with homogeneous Dirichlet boundary condition, where (−∆) s , s ∈ (0, 1), is the fractional Laplacian operator, λ ∈ R and ρ ∈ L ∞ (Ω). We study weak* continuity, convexity and Gâteaux differentiability of the map ρ → 1/λ 1 (ρ), where λ 1 (ρ) is the first positive eigenvalue. Moreover, denoting by G(ρ 0 ) the class of rearrangements of ρ 0 , we prove the existence of a minimizer of λ 1 (ρ) when ρ varies on G(ρ 0 ). Finally, we show that, if Ω is Steiner symmetric, then every minimizer shares the same symmetry.Recently, great attention has been focused on the study of fractional and nonlocal operators of elliptic type, both for pure mathematical research and in view of concrete real-world applications. This type of operators arises in a quite natural way in many different contexts, such as, among others, the thin obstacle problem,

Optimal location of resources and Steiner symmetry in a population dynamics model in heterogeneous environments

“…Indeed, as a matter of fact, every minimizer of λ 1 (m) in the class M has the form described in Theorem 1. Unfortunately, this does not follow from [9] as it happens for Theorem 1; instead, it can be shown reasoning exactly as in the proof of Theorem 1.1 in [1]. Due to the length of this argument, we prefer to include this topic in a future paper.…”

“…More precisely, we study the minimization of λ 1 (m) when m is chosen in an appropriate class of bounded measurable functions. Problem (1) originates from the study of reaction-diffusion equations in mathematical ecology which dates to the pioneering work [21] of Skellam. Precisely, we deal with the following model examined by Cantrell and Cosner in [5] and [6] (2)…”

“…Problem (1) with positive bounded weight m(x) also has a well known physical interpretation: it models a vibrating membrane Ω with clamped boundary ∂Ω and mass density m(x); the physical meaning of λ 1 (m) is the principal natural frequency of the membrane. Thus, the minimization of λ 1 (m) is physically equivalent to find the mass distribution of the membrane which gives the lowest principal natural frequency.…”

An attractive numerical algorithm for solving nonlinear Caputo–Fabrizio fractional Abel differential equation in a Hilbert space

“…Depending on the context, Abel differential equations of the first and second kinds are one of the most important nonlinear nonhomogeneous equations having a long history and various applications in physics, chemistry, biology, medicine, and epidemiology, including fuel mechanics, magnetic statistics, solid mechanics, thin film condensation, and medium problems [19,20]. For completeness, we also stress that Abel differential equations find applications in probability when full moment problems are involved and, moreover, partial and pseudodifferential equations also very suitably fit in the modeling of real phenomena like those aforementioned; see [21][22][23][24]. Furthermore, they are a generalization of the common Riccati and Bernoulli differential equations and of a particular class of logistic differential equations.…”

Maximization Problems for Second Order Elliptic Equations

“…Again in 2020, Emamizadeh et al investigated bang-bang and multiple valued optimal solutions of control problems related to quasi-linear elliptic equations [13]. In the same year, Anedda et al investigated the Steiner symmetry of the minimizer (in classes of rearrangements) of a fractional eigenvalue problem [14].…”