Mihalis Mourgoglou (Universidad de Pais Vasco)

Title: "Varopoulos' type extensions and applications to the solvability of Boundary Value Problems for second order elliptic operators in rough domains".

Abstract: In this talk I will talk about some recent advances in solvability of Boundary Value Problems for second order elliptic operators in rough domains. The main ingredient is the construction of Varopoulos' type extensions. Namely, those are extensions $u$ of $f \in L^p$ and $w$ of $g \in W^{1,p}$ boundary data so that the Carleson functional of $\nabla u$ and $Lw$ are in $L^p$ and the non-tangential maximal function of $u$ and $\nabla w$ are in $L^p$ with norms bounded by the $L^p$ norm of $f$ and the $ W^{1,p}$ norm of $g$ respectively. My plan is to show how those extensions appear in a natural way and sketch their construction.