30/11/2007, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Frank-Olme Speck, Instituto Superior Técnico, U.T. Lisboa
Boundary value problems for the Helmholtz equation in an octant
We consider a class of boundary value problems for the
three-dimensional Helmholtz equation that appears in diffraction
theory. On the three faces of the octant, which are quadrants, we
admit first order boundary conditions with constant coefficients,
linear combinations of Dirichlet, Neumann, impedance and/or oblique
derivative type. A new variety of surface potentials yields \(3
\times 3\) boundary pseudodifferential operators on the
quarter-plane that are equivalent to the operators associated to
the boundary value problems in a Sobolev space setting. These
operators are analyzed and inverted in particular cases, which
gives us the analytical solution of a number of well-posed
problems. The talk is based upon common research with Ernst
Stephan.