On low regularity problems: the elliptic and parabolic theory

In this talk I plan to present new solvability techniques for elliptic problems subject to boundary data assuming a minimal regularity property. These techniques heavily rely on certain harmonic analysis tools and recent progress in the theory of function spaces. As an illustration, I will discuss the solvability theory for the weakly harmonic map system into a compact manifold (which is a "borderline" problem). Applications in the parabolic setting will also be mentioned with a particular focus on the chemotaxis Navier-Stokes equations.