On a boundary value problem for pairs of polyanalytic functions (Q1042643)

The authors investigate the following boundary value problems. To find \(n\)-analytic functions \(F_1(z,\overline{z})=\sum_{k=0}^{n-1}\varphi_k(z)\overline{z}^k\) and \(F_2(z,\overline{z})=\sum_{k=0}^{n-1}\psi_k(z)\overline{z}^k\) in a finite simply-connected domain \(G\) satisfying on \(\Gamma:=\partial G\) one of the following three groups of conditions: (I) \(D_iF_1(t,{\overline{t}})=\overline{\widetilde{D}_iF_2(t,{\overline{t}})}+h_i(t)\), \(1\leq i\leq n\); (II) \(D_iF_1(\alpha_+(t),{\overline{\alpha_+(t)}})=\overline{\widetilde{D}_iF_2(t,{\overline{t}})}+h_i(t)\), \(1\leq i\leq n\); (III) \(D_iF_1(\alpha_-(t),{\overline{\alpha_-(t)}})=\overline{\widetilde{D}_iF_2(t,{\overline{t}})}+h_i(t)\), \(1\leq i\leq n\). Here \(\Gamma\) is a Lyapunov curve, \(h_i\) are given \(L_p\)-functions on \(\Gamma\), \[ D_{i}=\sum_{j=0}^{n-1} a_{ij}(t)\Delta_{s_j},\qquad {\widetilde{D}}_{i}=\sum_{j=0}^{n-1}a_{ij}(t)\Delta_{s_j}, \] \(a_{ij}\), \(b_{ij}\) are given continuous functions on \(\Gamma\), \[ 0\leq s_0<s_1+1<s_2+2<\dots<s_{n-1}+n-1, \] \[ 0\leq r_0<r_1+1<r_2+2<\dots<r_{n-1}+n-1, \] \[ \Delta_{s_j}=\partial^{s_j+j}/\partial t^{s_j}\partial \overline{t}^j,\qquad \Delta_{r_j}=\partial^{r_j+j}/\partial t^{r_j}\partial \overline{t}^j, \] \(\alpha_+\) and \(\alpha_-\) are two Carleman shifts of \(\Gamma\), where \(\alpha_+\) preserves, and \(\alpha_-\) reverses the orientation of \(\Gamma\). The problems (I) -- (III) are reduced to equivalent operator equations \(T_j\big(\Phi(t),\Psi(t)\big)=H(t)\), \(t\in \Gamma\), \(0\leq j\leq 2\), where \(\Phi\) and \(\Psi\) are the vector functions consisting of the analytic components \(\varphi_k \) and \(\psi_k\) of \(F_1\) and \(F_2\); \(H(t)=\big(h_0(t),\dots,h_{n-1}(t)\big)\). The perators \(T_j\) are considered as mappings from the product of some Sobolev spaces on \(\Gamma\) to \(L_p^n(\Gamma)\). The main theorem of the paper states that the operatros \(T_j\) are Fredholm operators iff \(\text{det}\,A(t)\neq 0\) and \(\text{det}\,B(t)\neq 0\), where \(A=\|a_{ij}\|_{0\leq i,j\leq n-1}\) and \(B=\|b_{ij}\|_{0\leq i,j\leq n-1}\). Under these conditions, \[ \text{ind}\,T_0=\text{ind}\,T_1=-1/(2\pi)\left[\text{arg}(\text{det}A(t)\text{det}B(t))\right]_{t\in \Gamma}+\sum_{k=0}^{n-1}(s_k+r_k)+2n, \] \[ \text{ind}\,T_2=1/(2\pi)\left[\text{arg}\,(\text{det}\,A(t)(\text{det}\,B(t))^{-1})\right]_{t\in \Gamma}+\sum_{k=0}^{n-1}(s_k+r_k)+2n. \]