2020/08/05 by Francesco Della Pietra, Carlo Nitsch, Della Pietra, Francesco +5 · 4 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations #Mathematical Approximation and Integration
paper · pdf · doi:10.48550/arxiv.2008.02193
We study thermal insulating of a bounded body \Ω\⊂ \ℝn.\nUnder a prescribed heat source f\≥ 0, we consider a model of heat transfer\nbetween \Ω and the environment determined by convection; this\ncorresponds, before insulation, to Robin boundary conditions. The body is then\nsurrounded by a layer of insulating material of thickness of size\n\ε>0, and whose conductivity is also proportional to \ε.\nThis corresponds to the case of a small amount of insulating material, with\nexcellent insulating properties. We then compute the \Γ-limit of the\nenergy functional F_\ε and prove that this is a functional F whose\nminimizers still satisfy an elliptic PDEs system with a non uniform Robin\nboundary condition depending on the distribution of insulating layer around\n\Ω. In a second step we study the maximization of heat content (which\nmeasures the goodness of the insulation) among all the possible distributions\nof insulating material with fixed mass, and prove an optimal upper bound in\nterms of geometric properties. Eventually we prove a conjecture which states\nthat the ball surrounded by a uniform distribution of insulating material\nmaximizes the heat content.\n