Weighted L^p theory for vector potential operators in three-dimensional exterior domains

Hela Louati (1) , Mohamed Meslameni (2) , Ulrich Razafison (3)

1 Laboratoire de Mathématiques de Sfax
2 Laboratoire de Mathématiques de Sfax
3 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)

In the present paper we study the vector potential problem in exterior domains of R^3. Our approach is based on the use of weighted spaces in order to describe the behaviour of functions at infinity. As a first step of the investigation, we prove important results on the Laplace equation in exterior domains with Dirichlet or Neumann boundary conditions. As a consequence of the obtained results on the vector potential problem, we establish usefull results on weighted Sobolev inequalities and Helmholtz decompositions of weighted spaces.

Mots clés: weighted spaces; Helmholtz decomposition; Vector potential; Sobolev inequalities; unbounded domains

Domaines: Equations aux dérivées partielles [math.AP]

Hela Louati, Mohamed Meslameni, Ulrich Razafison. Weighted L^p theory for vector potential operators in three-dimensional exterior domains. 2014. ⟨hal-01025281v2⟩