Boutet de Monvel Calculus

In this chapter we introduce the notion of transmission property due to Louis Boutet de Monvel (J. Anal. Math. 17:241–253, 255–304, 1966 and Acta Math. 126:11–51, 1971), which is a condition about symbols in the normal direction at the boundary. Elliptic boundary value problems cannot be treated directly by pseudo-differential operator methods. It was Boutet de Monvel who brought in the operator-algebraic aspect with his calculus in 1971. He constructed a relatively small “algebra”, called the Boutet de Monvel algebra, which contains the boundary value problems for elliptic differential operators as well as their parametrices. It should be emphasized that the Boutet de Monvel calculus is closely related to the classical Wiener–Hopf technique that remains an extremely important tool for modern scientists. In Sects. 9.1–9.4 we take a close look at Boutet de Monvel’s work. In Sect. 9.5, as an application of the Boutet de Monvel calculus we derive an index formula of Agranovič–Dynin type for the Neumann problem and the hypoelliptic Robin problem in the framework of \(L^{p}\) Sobolev spaces (see Theorem 9.27).