Modeling and Analysis of an Optimal Insulation Problem on Non-Smooth Domains
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arXiv
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2025
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| author | Antil, Harbir Kaltenbach, Alex Kirk, Keegan L. A. |
| author_facet | Antil, Harbir Kaltenbach, Alex Kirk, Keegan L. A. |
| contents | In this paper, we study an insulation problem that seeks the optimal distribution of a fixed amount $m>0$ of insulating material coating an insulated boundary $Γ_I\subseteq \partialΩ$ of a thermally conducting body $Ω\subseteq \mathbb{R}^d$, $d\in \mathbb{N}$. The thickness of the thin insulating layer $Σ_{I}^{\varepsilon}$ is given locally via $\varepsilon \mathtt{d}$, where $\mathtt{d}\colon Γ_{I}\to [0,+\infty)$ specifies the (to be determined) distribution of the insulating material. We establish $Γ(L^2(\mathbb{R}^d))$-convergence of the problem (as $\varepsilon\to 0^+$). Different from the existing literature, which predominantly assumes that the thermally conducting body $Ω$ has a $C^{1,1}$-boundary, we merely assume that $Γ_I$ is piece-wise flat. To overcome this lack of boundary regularity, we define the thin insulating boundary layer $Σ_{I}^{\varepsilon}$ using a Lipschitz continuous transversal vector field rather than the outward unit normal vector field. The piece-wise flatness condition on $Γ_I$ is only needed to prove the $\liminf$-estimate. In fact, for the $\limsup$-estimate is enough that the thermally conducting body $Ω$ has a $C^{0,1}$-boundary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_11903 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Modeling and Analysis of an Optimal Insulation Problem on Non-Smooth Domains Antil, Harbir Kaltenbach, Alex Kirk, Keegan L. A. Analysis of PDEs 35B40, 35J25, 35Q93, 49J45, 80A19 In this paper, we study an insulation problem that seeks the optimal distribution of a fixed amount $m>0$ of insulating material coating an insulated boundary $Γ_I\subseteq \partialΩ$ of a thermally conducting body $Ω\subseteq \mathbb{R}^d$, $d\in \mathbb{N}$. The thickness of the thin insulating layer $Σ_{I}^{\varepsilon}$ is given locally via $\varepsilon \mathtt{d}$, where $\mathtt{d}\colon Γ_{I}\to [0,+\infty)$ specifies the (to be determined) distribution of the insulating material. We establish $Γ(L^2(\mathbb{R}^d))$-convergence of the problem (as $\varepsilon\to 0^+$). Different from the existing literature, which predominantly assumes that the thermally conducting body $Ω$ has a $C^{1,1}$-boundary, we merely assume that $Γ_I$ is piece-wise flat. To overcome this lack of boundary regularity, we define the thin insulating boundary layer $Σ_{I}^{\varepsilon}$ using a Lipschitz continuous transversal vector field rather than the outward unit normal vector field. The piece-wise flatness condition on $Γ_I$ is only needed to prove the $\liminf$-estimate. In fact, for the $\limsup$-estimate is enough that the thermally conducting body $Ω$ has a $C^{0,1}$-boundary. |
| title | Modeling and Analysis of an Optimal Insulation Problem on Non-Smooth Domains |
| topic | Analysis of PDEs 35B40, 35J25, 35Q93, 49J45, 80A19 |
| url | https://arxiv.org/abs/2503.11903 |