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We study in detail Hodge–Helmholtz decompositions in nonsmooth exterior domains Ω⊂ℝ  N   filled with inhomogeneous and anisotropic media. We show decompositions of alternating differential forms of rank  q  belonging to the weighted L 2 ‐space L  s   2,  q  (Ω),  s ∈ℝ, into irrotational and solenoidal  q ‐forms. These decompositions are essential tools, for example, in electro‐magnetic theory for exterior domains. To the best of our knowledge, these decompositions in exterior domains with nonsmooth boundaries and inhomogeneous and anisotropic media are fully new results. In the Appendix, we translate our results to the classical framework of vector analysis  N =3 and  q =1, 2. Copyright © 2008 John Wiley & Sons, Ltd.

Hodge–Helmholtz decompositions of weighted Sobolev spaces in irregular exterior domains with inhomogeneous and anisotropic media