Mathematical analysis of a partial differential equation system on the thickness

This study focuses on linear partial differential equation (PDE) systems that arise in topology optimization where the thickness of a structure is constrained. The thickness derived from the PDE is a fictitious one, and the key challenge of this work is to verify its equivalence to the intuitive, geometrically defined thickness. The main difficulty lies in that while intuitive thickness is determined solely by the shape, the thickness defined by the PDE depends not only on the shape but also on the entire domain and the diffusion coefficients used in solving the PDE. In this paper, we demonstrate that the thickness of an infinite, straight film as a simple shape with constant thickness is equivalent within a general domain. The proof involves constructing a reference solution within a special domain and evaluating the difference using the maximum (modulus) principle and an interior $H^1$ estimate. Additionally, we provide an estimate of the dependence of thickness on the diffusion coefficient.