Boundary value problems for elliptic systems with $L2$ data

Abstract

We describe joint work with Andreas Axelsson. We introduce a setup to study solutions of divergence form second order elliptic systems $Lu=0$ on the upper-half space with coefficients having little smoothness on the vertical variable (A Carleson measure estimate introduced by Dahlberg: this estimate being zero meaning that the coefficients do not depend on the vertical variable). In this setup, natural trace spaces on the boundary allow to describe limiting behaviour of our weak solutions $u$ and then to formulate and study Dirichlet, Neumann and Dirichlet-regularity problems. We obtain a perturbation result for well-posedness for small Carleson measures. This yield well-posedness results for a class of elliptic systems extending the known results of Dahlberg and Kenig-Pipher for real symmetric equations. Methods no longer use harmonic measure (which is unavailable) but a first order system formalism in the conormal gradient, the solution of the Kato conjecture and new maximal regularity estimates.