Bounded L-index in joint variables and analytic solutions of some systems of PDE's in bidisc

Анотація

The linear higher-order systems of PDE's with analytic coefficients in bidisc are considered. We study boundedness of the $\mathit{L}$-index in joint variables of their analytic solutions, where $\mathit{L}(z_1,z_2)=\left(l_1(z_1,z_2), l_2(z_1,z_2)\right)$, $l_j:{\mathbb{D}}^2\to {\mathrm{ℝ}}_{+}$  is continuous function, $j\in \left\{1,2\right\}$, ${\mathbb{D}}^2=\left\{\left(z_1,z_2\right)\in \mathrm{ℂ} : \left|z_1\right|<1, \left|z_2\right|<1\right\}$. The main tool of investigations is known Hayman's theorem. Some analytic solutions in bidisc for these system of PDE's are given.