ИСТИНА

Unique solvability of an exterior Dirichlet problem implies existence of an operator that mapsthe Dirichlet data (function on the obstacle boundary) to the normal derivative of the solution (anotherfunction on the boundary). The so defined Dirichlet-to-Neumann map DtN is a boundary pseudodifferential operator of order 1. In 2D problems, the boundary is one-dimensional, usually diffeomorfic to a circle, andDtN can be exactly (without truncation by order) described by a discrete symbol, which is a function ofthree parameters: boundary parameter s, Fourier series index (discrete momentum) n, and the wavenumberk. As k goes to infinity, the symbol has a nice asymptotic behaviour uniformly in s and n. This fact can beviewed as a microlocal refinement of the Kirchhoff approximation. A related numerical method will alsobe discussed.