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Article type: Research Article
Authors: Bejaoui, Eyaa; * | Ben Belgacem, Fakerb
Affiliations: [a] Université Tunis El Manar, ENIT-LAMSIN, B.P. 37, 1002 Tunis-Belvédère, Tunisie | [b] Alliance Sorbonne Université, UTC, EA 2222, Laboratoire de Mathématiques Appliquées de Compiègne, F-60205 Compiègne, France
Correspondence: [*] Corresponding author. E-mail: eya.bejaoui_02@utc.fr.
Abstract: We explore the mathematical structure of the solution to an elliptic diffusion problem with point-wise Dirac sources. The conductivity parameter is space-varying, may have jumps and the Dirac sources may be located along the discontinuity curves of that parameter. The variational problem, issued by duality, is proven to be well posed using a sharp elliptic regularity result by Di-Giorgi [Mem. Accad. Sci. Torino, 3, 1957]. The paper is aimed at a key expansion into a split singular/regular contributions. The singular part is calculated by an explicit formula, while the regular correction can be computed as the solution to a standard variational Poisson problem. The latter can be successfully approximated by most of the numerical methods practiced nowadays. Some analytical examples are discussed at last to assess the minimality of the assumptions we use to establish our theoretical results.
Keywords: Elliptic Problems, point-wise Dirac sources, non-smooth conductivity, singular/regular splitting
DOI: 10.3233/ASY-221824
Journal: Asymptotic Analysis, vol. 134, no. 1-2, pp. 1-23, 2023
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