Speaker
Description
We consider strongly elliptic second-order systems with constant coefficients in domains on the plane reduced to the canonical form with only two real parameters $\tau$, $\sigma\in[0, 1)$ and written in complex form. When both parameters are equal to zero, we obtain the complex Laplace equation. When only $\sigma=0$, we have the skew-symmetric system which is a perturbation of the Laplace one by the parameter $\tau$. Its solutions are represented as a sum of two holomorphic functions of variables $z_\tau=z-\tau\overline z$ and $\overline z$. In some regions, it is possible to construct the Green's function for a skew-symmetric system using the analogue of Schwarz function which holomorphically expresses $z_\tau$ on the boundary in terms of $\overline z$. Although different Schwarz functions may arise for different parts of the boundary, for such domains as, for example, a strip or an angle, it is possible to select functions that are invariant under the replacement of one Schwarz function by another. Using such functions, the Green's function is constructed. The general case of a two-parameter system is studied by considering it as a perturbation of a skew-symmetric system with respect to the parameter $\sigma$.
