Numerical studies of the optimization of the first eigenvalue of the heat diffusion in inhomogeneous media

Abstract

In this paper, we study optimization of the first eigenvalue of −∇ · (ρ(x)∇u) = λu in a bounded domain Ω ⊂ R n under several constraints for the function ρ. We consider this problem in various boundary conditions and various topologies of domains. As a result, we numerically observe several common criteria for ρ for optimizing eigenvalues in terms of corresponding eigenfunctions, which are independent of topology of domains and boundary conditions. Geometric characterizations of optimizers are also numerically observed.1. The element ρ * ∈ K optimizing λ 1 (ρ) can be characterized by inequalities with respect to ρ * ∇u * , where u * is the eigenfunction associated with λ 1 (ρ * ) of (1.1). As a consequence, the domain S * = {x ∈ Ω | ρ * (x) = c} is given by the super-or the sub-level set of |ρ * ∇u * |. This characterization is independent of topology and geometry of Ω and boundary conditions on ∂Ω.2. If Ω is star-shaped and symmetric in a certain direction, then S * has the same symmetry.3. Optimized region S * depends continuously on a parameter of the boundary condition if the Robin boundary condition is imposed.The precise statements are described in Section 4 (Observation 4.2 -4.4) with various numerical results. This paper is organized as follows. In Section 2, we provide more precise setting of our problems. A numerical method for finding optimizers we apply here, the level set approach, is also derived here. In Section 3, numerical and mathematical known results for a well-considered problem are discussed. Section 4 is where our main discussion is developed. We show several numerical observations about eigenvalue optimization criteria, geometry of the level set S * = {x ∈ Ω | ρ * (x) = c} for the optimizer ρ * and continuous dependence of S * on boundary conditions. 2 Setting 2.