Pré-Publication, Document De Travail Année : 2023

Singularity Removal for 3D Elliptic Problems with Variable Coefficients and Line Sources

Résumé

Three dimensional elliptic problems with variable coefficients and line Dirac sources arise in a number of fields. The lack of regularity on the solution prompts users to turn towards alternative variational formulations. Rather than using weighted Sobolev spaces, we prefer the dual variational formulation written in the Hilbertian Lebesgue space, the one used by G. Stampacchia [Séminaire Jean Leray, 1964]. The key work is to show a singular/regular expansion where the singularity of the potential is fully expressed by a convolution formula, based on the Green kernel of the Laplacian. The correction term restores the boundary condition and fits with the standard variational formulation of Poisson equation (in the Sobolev space H^1). We intend to develop a thorough analysis of the proposed expansion while avoiding stringent assumptions on the conductivities. Sharp technical tools, as those developed in [E. Di-Giorgi, Mem. Accad. Sci. Torino. 1957] and [N. G. Meyers Ann. Scuo. Norm. Sup. Pisa, 1963], are necessary in the proofs.
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Dates et versions

hal-04337080 , version 1 (12-12-2023)

Identifiants

  • HAL Id : hal-04337080 , version 1

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Eya Bejaoui, Faker Ben Belgacem. Singularity Removal for 3D Elliptic Problems with Variable Coefficients and Line Sources. 2023. ⟨hal-04337080⟩
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