Variational approach in weighted Sobolev spaces to scattering by unbounded rough surfaces

We consider the problem of scattering of time harmonic acoustic waves by an unbounded sound soft surface which is assumed to lie within a finite distance of some plane. The paper is concerned with the study of an equivalent variational formulation of this problem set in a scale of weighted Sobolev spaces. We prove well-posedness of this variational formulation in an energy space with weights which extends previous results in the unweighted setting [S. Chandler-Wilde and P. Monk, SIAM J. Math. Anal., 37 (2005), pp. 598–618] to more general inhomogeneous terms in the Helmholtz equation. In particular, in the two-dimensional case, our approach covers the problem of plane wave incidence, whereas in the three-dimensional case, incident spherical and cylindrical waves can be treated. As a further application of our results, we analyze a finite section type approximation, whereby the variational problem posed on an infinite layer is approximated by a variational problem on a bounded region.

The growth of private property vehicles in the UK: causes and conditions

The UK private indirect real estate market has seen a rapid growth in the last seven years. The gross asset value (GAV) of the private property vehicle (PPV) market has about tripled from a GAV of £22.6bn in 1998 to a GAV of £67.1 billion at the end of 2005 (OPC, 2006). Although this trend of growing syndication of real estate is not only a UK phenomenon, the rate of growth has been significantly faster in the UK. For example the German open-ended funds have grown over the same period from €50.4bn to €85.1bn (BVI, 2006). In the US the market capitalization of equity real estate investment trusts (REIT) has grown 155% since 1999 to US$ 301bn (NAREIT, 2006). Each jurisdiction is offering different formats to invest indirectly into real estate but at the core all these vehicles are the same in that they provide a different route for investors to access real estate. In the UK, although the range of ‘products’ is now quite diverse, all structures have in common the ‘wrapping’ of property assets into a multi-investor vehicle. This paper examines the nature, pattern and process of market growth in PPVs and constructs a series of associations between causes and effects to explain this market shift.