Lucia Swoboda, Chalmers/Göteborgs universitet

Abstract: In the theory of partial differential equations, it is often useful to consider transformations on certain function spaces. The modified Hilbert transform is one such transformation. It arises naturally when deriving a Galerkin-Bubnov variational formulation for the heat equation in anisotropic Sobolev spaces and leads to a symmetric and positive definite form. In this talk, I will introduce the modified Hilbert transform and discuss its main properties. I will explain how this operator leads to unique solvability of a variational formulation for the heat equation in anisotropic Sobolev spaces, and to stability in the case of space-time tensor product discretization. I will also present some ideas for possible extensions of this theory.