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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