A strange term coming from the boundary data

2022/10/03 by Aaron Pim, Pim, Aaron
Computer Science · Engineering · Mathematics · #35B27 #74Q15 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Composite Material Mechanics #FOS: Mathematics #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2210.00820

openalex publication_date 2022/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, I derive the limiting behaviour of the solutions to Poisson's equation, in a perforated domain, subject to inhomogeneous Robin boundary conditions. In the first half of the paper, I derive a generalised limit for non-periodic domains and arbitrary boundary data. In the second half of this paper, I demonstrate that for periodically arranged spheres and identical Robin boundary data on each sphere, the homogenised limit of Poisson's equation satisfies the Helmholtz equation with an additional term in the domain data, which represents the contribution from the inhomogeneous Robin boundary data. These results are a generalisation of the work of Kaizu, who derived the limit of the solutions to the homogeneous Robin problem.

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