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Dr. Giles Auchmuty
Professor of Mathematics
University of Houston
This talk will describe the use of elementary Hilbert space methods to prove results about solutions of boundary value problems for Laplace’s equation. Results about the subspaces of real harmonic functions as subspaces of the Hilbert spaces L2(Ω) and Hm(Ω) will be described. Boundary value problems for Laplace’s equation may be viewed as studying the linear mapping of some space of allowable boundary data to these Hilbert spaces.
Under natural conditions on the boundary, these are compact linear transformations that have a singular value decomposition (SVD). This SVD will be described in terms of Steklov eigen values and eigen functions and their use for efficient approximations of solutions of different boundary value problems will be illustrated. These representations of the solutions are related to applications such as pipe flow and electrostatic fields.