Reflection Principle for Systems of First Order Elliptic Equations with Analytic Coefficients

Abstract

Let T be a simply connected domain of the $z = x + iy$ plane, whose boundary contains a portion $\sigma$ of the x-axis. Also let $A(z,\zeta ),B(z,\zeta ),F(z,\zeta ),\alpha (z),\beta (z)$ and $\rho (z)$ be holomorphic functions for $z,\zeta \in T \cup \sigma \cup \bar T$, with $\alpha (z) - i\beta (z) \ne 0$ for $z \in \bar T \cup \sigma ,\alpha (z) + i\beta (z) \ne 0$ for $z \in T \cup \sigma$. Furthermore, we assume that $\alpha (x)$ and $\beta (x)$ are real valued functions for $x \in \sigma$. Our reflection principle states that for any solution $w = u + iv$ of an equation of the type $\partial w/\partial \bar z = A(z,\bar z)w + B(z,\bar z)\bar w + F(z,\bar z)$ in T under the boundary condition $\alpha (x)u + \beta (x)v = \rho (x)$ on $\sigma ,w$ can be continued analytically across the x-axis, onto the entire mirror image $\bar T$.