Riemann–Hilbert Problem for First-Order Elliptic Systems with Constant Leading Coefficients on the Plane

In a finite domain $D$ of the complex plane bounded by a smooth contour $\Gamma$, we consider the Riemann–Hilbert boundary-value problem
\begin{equation*} \operatorname{Re} CU^+=f \end{equation*}
for the first-order elliptic system
\begin{equation*} \frac{\partial U}{\partial y}-A\frac{\partial U}{\partial x}+a(z)U(z)+b(z)\overline{U(z)}=F(z) \end{equation*}
with constant leading coefficients. Here $+$ denotes the boundary value of the function $U$ on $\Gamma$, the constant matrices $A_1, A_2 \in\mathbb{C}^{l\times l}$ and $(l\times l)$-matrix coefficients $a$ and $b$ belong to the Hölder class $C^{\mu}$, $0<\mu<1$, and $(l\times l)$-matrix function $C$ belongs to the class $C^\mu(\Gamma)$. We prove that in the class $U\in C^\mu(\overline{D})\cap C^1(D)$, this problem is a Fredholm problem and its index is given by the formula
\begin{equation*} \varkappa=-\sum_{j=1}^m\frac{1}{\pi} \big[\arg\det G\big]_{\Gamma_j}+(2-m)l. \end{equation*}