On the placement of an obstacle so as to optimize the Dirichlet heat content

2021/06/23 by Liangpan Li, Li, Liangpan
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Boundary (topology) #Boundary value problem #Bounded function #Combinatorics #Content (measure theory) #Dirichlet distribution #Domain (mathematical analysis) #FOS: Mathematics #Geography #Geometry #Mathematical Approximation and Integration #Mathematical analysis #Mathematics #Nonlinear Partial Differential Equations #Obstacle #Obstacle problem #Physics #Regular polygon #SPHERES #Spectral Theory (math.SP) #math.SP

paper · pdf · doi:10.48550/arxiv.2106.12480

arxiv created 2021/06/23 · openalex publication_date 2021/06/23 · arxiv updated 2021/06/24 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

We prove that among all doubly connected domains of Rⁿ (n>=2) bounded by two spheres of given radii, the Dirichlet heat content at any fixed time achieves its minimum when the spheres are concentric. This is shown to be a special case of a more general theorem concerning the optimal placement of a convex obstacle inside some larger domain so as to maximize or minimize the Dirichlet heat content.

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