Regularity of Weak Solutions to Linear Parabolic Systems with Measurable Coefficients

Introductory Talk: We give an introduction to the theory of (complex) interpolation of Banach spaces and give a sketch of the proof the ubiquitous Snieberg lemma. This lemma roughly states that one can extrapolate invertibility for linear operators which are bounded on a complex interpolation scale. The only tools required here are basic complex analysis and functional analysis.

Main Talk: We use the ideas of the first talk to study the regularity of solutions to some (linear) partial differential equations, with a particular focus on linear parabolic systems with (complex) coefficients satisfying the "weak Garding inequality" (ellipticity-type condition).