Modeling and Analysis of an Optimal Insulation Problem on Nonsmooth Domains
Harbir Antil, Alex Kaltenbach, Keegan L. A. Kirk · SIAM Journal on Mathematical Analysis · 2026
Abstract. In this paper, we study an insulation problem that seeks the optimal distribution of a fixed amount [Formula: see text] of insulating material coating an insulated boundary [Formula: see text] of a thermally conducting body [Formula: see text], [Formula: see text]. The thickness of the thin insulating layer [Formula: see text] is given locally via [Formula: see text], where [Formula: see text] specifies the (to be determined) distribution of the insulating material. We establish [Formula: see text]-convergence of the problem (as [Formula: see text]). Different from the existing literature, which predominantly assumes that the thermally conducting body [Formula: see text] has a [Formula: see text]-boundary, we merely assume that [Formula: see text] is piecewise flat. To overcome this lack of boundary regularity, we define the thin insulating layer [Formula: see text] using a Lipschitz continuous (globally) transversal vector field rather than the outward unit normal vector field. The piecewise flatness condition on [Formula: see text] is only needed to prove the [Formula: see text]-estimate. In fact, for the [Formula: see text]-estimate it is enough that the thermally conducting body [Formula: see text] has a [Formula: see text]-boundary.