Optimization of the first eigenvalue of the heat diffusion in inhomogeneous media: Global well-posedness of the viscous approximation problems (Topology optimization theory and applications toward wide fields of natural sciences)

Abstract

We consider the optimization of the first eigenvalue of -nabla. ( $rho$nabla u) = $lambda$ u on a bounded domain $Omega$ subset mathbb{R}^{n} with a constraint on the diffusion coefficient $rho$. We reduce our optimization problem to the Hamilton-Jacobi type equation for the function determining $rho$ via the level set method. We take the viscous approximation of the Hamilton-Jacobi type equation and prove its global well-posedness. We expect that solutions of the original Hamilton-Jacobi type equation are obtained as the limit of its viscous approximation. We also expect that the proposing approach leads to the optimization analysis of general energy functionals including constraints. This article is written in Japanese.