Discontinuous boundary value problems for first order systems of mixed type (Q2751176)

In this paper the system of mixed elliptic-hyperbolic type NEWLINE\[NEWLINEu_x - v_y = au + bv + f, \quad v_x + (\text{sgn} y) \cdot u_x = cu + dv + gNEWLINE\]NEWLINE in domain \(D\) with boundary consisting for \(y>0\) of arc of a single radius circle with endpoints \(A=(0,0)\) and \(B=(2,0)\) and for \(y<0\) of characteristics \(AC\) and \(CB\) of system are considered. The problem with discontinuous boundary conditions NEWLINE\[NEWLINE[\lambda_1(z) u(z) - \lambda_2(z) v(z)] \Bigl|_{z=z_j} = s_j,\quad z\in \{z_i \mid z_i\in AC\cup BC,\;i=\overline{0,n}\},NEWLINE\]NEWLINE NEWLINE\[NEWLINE\left.z_j=\begin{cases} x_j/2- x_j/2i,& j\text{ is odd number}\\ 1+x_j/2+ (x_j/2-1)i, &j \text{ is even number}\end{cases}\right\}, \quad 0=x_0<x_1< \ldots < x_n < x_{n+1}=2;NEWLINE\]NEWLINE NEWLINE\[NEWLINE \lambda_1(z) u(z) + \lambda_2(z) v(z) = r(z),\quad z \in \partial D \setminus \{z_i\}, NEWLINE\]NEWLINE where \(s_j\) are given numbers, \(\lambda_1,\) \(\lambda_2,\) \(r\) are given functions satisfying certain conditions are researched. It is proved that this problem has a unique solution.NEWLINENEWLINEFor the entire collection see [Zbl 0966.00026].