Seminar

Stability of some Fractional function spaces

Fractional Sobolev spaces, which generalize the concept of classical Sobolev spaces, have been a central tool in harmonic analysis, variation problems, and PDE for several decades. One of the main issues of these spaces is that they are unstable as the fractional parameter s goes to 1. In this talk, I will share some of the new results that I've obtained regarding stabilizing these spaces, and how these results compare to the classical Bourgain-Brezis- Mironescu formula.

A new solvability condition for $L^p$ boundary value problems

Abstract: We are discussing the elliptic operator $L:=\mathrm{div}(A\nabla\cdot)$ and wonder which types of matrices $A$ yield solvability of $L^p$ boundary value problems. It is well-known that the DKP or Carleson condition implies solvability for the Dirichlet and the regularity boundary value problem. Equally, if the domain is the upper half space, independence of the transversal direction $t$ gives solvability of these boundary value problems.