A uniqueness theorem for linear elliptic equations with dominating derivative with respect to \(\bar{z}\) (Q323580)

The author is interested in uniqueness results for solutions to the elliptic equation \[ \partial^n_{\bar{z}} u + \sum_{0 \leq k+j \leq n-1} A_{kj}(z)\partial^{k}_z \partial^j_{\bar{z}} u=0, \] where the coefficients \(A_{kj}=A_{kj}(z)\) are supposed to be analytic in \(x\) and \(y\), \(z=x+iy\), in a domain \(G \subset \mathbb{C}\). A question of interest is a zero set \(M \subset G\) for solutions \(u=u(z)\), that is, if \(u \equiv 0\) on \(M\), then \(u \equiv 0\) in \(G\). It is known that zero sets for holomorphic functions (solutions of \(\partial_{\bar{z}} u=0\)) and for harmonic functions (solutions of \(\Delta u=0\)) are quite different. To describe zero sets the author uses ideas from the monograph [\textit{M. B. Balk}, Polyanalytic functions. Berlin: Akademie-Verlag (1991; Zbl 0764.30038)]. There the notion of an accumulation point of order \(n\) of zeros of a given function is introduced. The author proves the following result: If \(z_0 \in G\) is an accumulation point of order \(n\) of zeros of a given classical solution \(u\) to the above elliptic equation, then \(u \equiv 0\) in \(G\).