{"ID":2828853,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.13098","arxiv_id":"2512.13098","title":"Convection Effects and Optimal Insulation: Modelling and Analysis","abstract":"In this paper, we study an insulation problem that seeks to determine the optimal distribution of a given amount $m\u003e0$ of insulating material coating an insulated boundary part $Γ_I\\subseteq \\partialΩ$ of a thermally conducting body $Ω\\subseteq \\mathbb{R}^d$, $d\\in \\mathbb{N}$, subject to convective heat transfer. The `$\\textit{thickness}$' of the insulating layer $Σ_{I}^{\\varepsilon}\\subseteq \\mathbb{R}^d$ is given locally via $\\varepsilon \\mathtt{d}$, where $\\varepsilon\u003e0$ denotes the (arbitrarily small) conductivity and $\\mathtt{d}\\colon Γ_{I}\\to [0,+\\infty)$ the (to be determined) distribution of the insulating material. Then, the physical process is modelled by the stationary heat equation in the insulated thermally conducting body $Ω_{I}^{\\varepsilon}:= Ω\\cupΣ_{I}^{\\varepsilon}$ with Robin-type boundary conditions on the interacting insulation boundary $Γ_I^{\\varepsilon}\\subseteq \\partialΩ_{I}^{\\varepsilon}$ (reflecting convective heat transfer between the thermally conducting body $Ω$ and its surrounding medium) as well as Dirichlet and Neumann boundary conditions at the remaining boundary parts, $\\textit{i.e.}$, $\\partialΩ_{I}^{\\varepsilon}\\setminus Γ_I^{\\varepsilon}$. More precisely, we establish $Γ(L^2(\\mathbb{R}^d))$-convergence of the heat loss formulation (as ${\\varepsilon \\to 0^+}$), in the case that the thermally conducting body $Ω$ is a bounded Lipschitz domain having a $C^{1,1}$-regular or piece-wise flat insulated boundary $Γ_I$.","short_abstract":"In this paper, we study an insulation problem that seeks to determine the optimal distribution of a given amount $m\u003e0$ of insulating material coating an insulated boundary part $Γ_I\\subseteq \\partialΩ$ of a thermally conducting body $Ω\\subseteq \\mathbb{R}^d$, $d\\in \\mathbb{N}$, subject to convective heat transfer. The...","url_abs":"https://arxiv.org/abs/2512.13098","url_pdf":"https://arxiv.org/pdf/2512.13098v1","authors":"[\"Harbir Antil\",\"Alex Kaltenbach\",\"Keegan L. A. Kirk\"]","published":"2025-12-15T08:56:22Z","proceeding":"math.AP","tasks":"[\"math.AP\",\"math.OC\"]","methods":"[]","has_code":false}