Nonlinear elliptic PDEs subject to irregular data

In this talk, I will discuss a new approach to the solvability of semilinear elliptic problems (in bounded and unbounded domains) supplemented by data assuming low regularity. The method exploits the nature of the nonlinearity as well as other intrinsic properties of the equation to select suitable functional frameworks where solutions are sought for. In the process, we borrow tools from modern harmonic analysis and function spaces theory.

Two equations of interest will be mentioned: The weakly harmonic map problem and the stationary Navier-Stokes flow. I also plan to comment on the parabolic theory if time allows.