An Optimization Problem in Heat Conduction With Volume Constraint and Double Obstacles

2022/01/21 by Xiaoliang Li, Cong Wang, Li, Xiaoliang +1
Computer Science · Engineering · Mathematics · #35J20 #35R35 #49N60 #80A22 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Topology Optimization in Engineering #math.AP #msc:35J20 #msc:35R35 #msc:49N60 #msc:80A22

paper · pdf · doi:10.48550/arxiv.2201.08587

20 pages

arxiv created 2022/01/21 · openalex publication_date 2022/01/21 · arxiv updated 2022/01/24 · openalex created_date 2023/01/04 · openalex updated_date 2026/07/28

Abstract

We consider the optimization problem of minimizing ∫_ℝⁿ|∇ u|² dx with double obstacles ϕ≤ u≤ψ a.e. in D and a constraint on the volume of \u>0\∖D, where D⊂ℝⁿ is a bounded domain. By studying a penalization problem that achieves the constrained volume for small values of penalization parameter, we prove that every minimizer is C^1,1 locally in D and Lipschitz continuous in ℝⁿ and that the free boundary ∂\u>0\∖D is smooth. Moreover, when the boundary of D has a plane portion, we show that the minimizer is C^1,(1)/(2) up to the plane portion.

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