Dirichlet problem for sets with higher co-dimensional boundaries.


For a given bounded domain $\Omega$ with non-tengential access to its boundary, the Dirichlet problem (for the Laplacian) is solvable if and only if the boundary is uniformly rectifiable. In this talk, I shall first present this nice result linking geometry and PDE, and we will discuss the literature surrounding it. Together with Guy David and Svitlana Mayboroda, we aim to extend this characterization of uniform rectifiability to the low dimension. I will talk about our strategy, the results that we successfully extended in low dimension, as well as the differences with the above classical case.