A boundary problem for an elliptic system of differential equations involving a discontinuous weight

✍ Scribed by M. Faierman

• Publisher: John Wiley and Sons
• Year: 2009
• Tongue: English
• Weight: 215 KB
• Volume: 282
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200710744

✦ Synopsis

We consider a boundary problem for an elliptic system of differential equations in a bounded region Ω ⊂ R n and where the spectral parameter is multiplied by a discontinuous weight function ω(x) = diag(ω1(x), . . . , ωN (x)). The problem is considered under limited smoothness assumptions and under an ellipticity with parameter condition. In a recent paper, Denk, Faierman, and Möller have studied this problem under the assumption that the ωj(x) -1 are essentially bounded in Ω. In this paper we suppose that ω(x) vanishes identically in a proper subregion ΩN 0 of Ω and that the ωj(x) -1 's are essentially bounded in Ω = Ω\ΩN 0 . By appealing to a result of Faierman pertaining to the Calderón method of reducing a boundary problem for a region to a problem on its boundary, we are able remove the region ΩN 0 altogether and reduce our boundary problem to one for the region Ω with a special type of boundary condition on ΓN 0 . Since this latter problem falls into a class of problems studied in the paper by Denk et al. cited above, all the spectral theory derived in that paper carries over directly to the problem which we are presently considering. Thus in this way we establish the spectral theory for the boundary problem under consideration here.