2008/05/04 by Guillaume Bal, Bal, Guillaume, Olivier Pinaud +1
Computer Science · Engineering · Mathematics · #35B40 #35J15 #35R30 #65R20 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #math.AP #msc:35B40 #msc:35J15 #msc:35R30 #msc:65R20
paper · pdf · doi:10.48550/arxiv.0805.0429
42 pages
arxiv created 2008/05/04 · openalex publication_date 2008/05/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper analyzes the influence of general, small volume, inclusions on the trace at the domain's boundary of the solution to elliptic equations of the form ∇ ⋅ D^\eps ∇ u^\eps=0 or (-Δ+ q^\eps) u^\eps=0 with prescribed Neumann conditions. The theory is well-known when the constitutive parameters in the elliptic equation assume the values of different and smooth functions in the background and inside the inclusions. We generalize the results to the case of arbitrary, and thus possibly rapid, fluctuations of the parameters inside the inclusion and obtain expansions of the trace of the solution at the domain's boundary up to an order \eps2d, where d is dimension and \eps is the diameter of the inclusion. We construct inclusions whose leading influence is of order at most \epsd+1 rather than the expected \epsd. We also compare the expansions for the diffusion and Helmholtz equation and their relationship via the classical Liouville change of variables.