Functional Calculus for First Order Systems of Dirac Type and Boundary Value Problems

Résumé du livre

It was shown recently that solutions of boundary value problems for some second order elliptic equations (or systems) in divergence form with measurable coefficients can be constructed from solutions of generalised Cauchy-Riemann systems, in the spirit of what can be done for the Laplace equation. This involves a first order bisectorial operator of Dirac type on the boundary whose bounded holomorphic functional calculus on $L^2$ is proved by techniques from the solution of the Kato problem, and the system can henceforth be solved by a semigroup for $L^2$ data in a spectral space. This memoir investigates the properties of this semigroup and, more generally, of the functional calculus on other spaces: $L^p$ for $p$ near 2 and adapted Hardy spaces otherwise. This yields non-tangential maximal functions estimates and Lusin area estimates for solutions of the boundary value problems.