An index formula for elliptic systems in the plane

Abstract

An index formula is proved for elliptic systems of P.D.E.’s with boundary values in a simply connected region Ω \Omega in the plane. Let A \mathcal {A} denote the elliptic operator and B \mathcal {B} the boundary operator. In an earlier paper by the author, the algebraic condition for the Fredholm property, i.e. the Lopatinskii condition, was reformulated as follows. On the boundary, a square matrix function Δ B + \Delta ^{+}_{{\mathcal {B}}} defined on the unit cotangent bundle of ∂ Ω \partial \Omega was constructed from the principal symbols of the coefficients of the boundary operator and a spectral pair for the family of matrix polynomials associated with the principal symbol of the elliptic operator. The Lopatinskii condition is equivalent to the condition that the function Δ B + \Delta ^{+}_{{\mathcal {B}}} have invertible values. In the present paper, the index of ( A , B ) ({\mathcal {A}},{\mathcal {B}}) is expressed in terms of the winding number of the determinant of Δ B + \Delta ^{+}_{{\mathcal {B}}} .