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 $\Omega$ in the plane. Let $\mathcal {A}$ denote the elliptic operator and $\mathcal {B}$ 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 $\Delta ^{+}_{{\mathcal {B}}}$ defined on the unit cotangent bundle of $\partial \Omega$ 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 $\Delta ^{+}_{{\mathcal {B}}}$ have invertible values. In the present paper, the index of $({\mathcal {A}},{\mathcal {B}})$ is expressed in terms of the winding number of the determinant of $\Delta ^{+}_{{\mathcal {B}}}$.