*preprint*

**Inserted:** 5 jul 2024

**Year:** 2023

**Abstract:**

We are interested in the thermal insulation of a bounded open set $\Omega$ surrounded by a set whose thickness is locally described by $\varepsilon h$, where $h$ is a non-negative function defined on the boundary $\partial\Omega$. We study the problem in the limit for $\varepsilon$ going to zero using a first-order asymptotic development by $\Gamma$-convergence.