Pseudodifferential Methods for Boundary Value Problems

In these lecture notes we introduce some of the concepts and results from microlocal analysis used in the analysis of boundary value problems for elliptic differential operators, with a special emphasis on Dirac-like operators. We first consider the problem of finding elliptic boundary conditions for the ∂̄operator on the unit disk. The rather explicit results in this special case delineate the route we follow for general first order elliptic systems on manifolds with boundary. After some geometric and functional analytic preliminaries, needed to do analysis on manifolds with boundary, we define and describe pseudodifferential operators satisfying the transmission condition. These operators behave well on data with support in a compact subset with a smooth boundary, and include the fundamental solutions of elliptic differential operators. Using the fundamental solution, we define the Calderon projector and establish its basic properties. We then consider boundary conditions defined by pseudodifferential projections, and find a simple criterion for such a boundary operator to define a Fredholm problem. This includes standard elliptic boundary conditions, as well as certain subelliptic problems. A formula is given for the index of such a boundary value problem in terms of the relative index between projectors on the boundary. 1991 Mathematics Subject Classification. 58J32, 30E125, 35F15, 35J55, 47A53;