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We deal with Oseen’s equations in the whole space. A class of existence, uniqueness and regularity results for both the scalar and the vectorial equations are given. Isotropic weighted Sobolev spaces are used for describing the growth or the decay of functions at infinity.
The Dirichlet problem and the construction of superharmonic functions with point harmonic singularities are two of the basic problems in potential theory. In this article, we study these problems in the context of discrete potential theory, which leads to the consideration of Green’s formulas and flux on a Cartier tree.
This paper gives applications of the enclosure method introduced by the author to typical inverse obstacle and crack scattering problems in two dimensions. Explicit extraction formulae of the convex hull of unknown polygonal sound-hard obstacles and piecewise linear cracks from the far field pattern of the scattered field at a fixed wave number and at most two incident directions are given. The main new points of this paper are: a combination of the enclosure method and the Herglotz wave function; explicit construction of the density in the Herglotz wave function by using the idea of the Vekua transform. By virtue of the construction, one can avoid any restriction on the wave number in the extraction formulae. An attempt for the case when the far field pattern is given on limited angles is also given.