A sign-changing Poisson kernel for a non-symmetric elliptic operator in a bounded domain

Abstract

We study the Dirichlet problem in the unit disk for a uniformly elliptic divergence-form operator whose skew-symmetric part has a jump discontinuity and is controlled by a real parameter $k$. Using a first-order Dirac systems method, we obtain explicit solution formulas, $L^2$ non-tangential maximal estimates, and almost-everywhere convergence to the prescribed boundary data. We show that the associated $L^2$ boundary equation undergoes a sharp transition at $|k|=1$, reflected in three natural $L^2$ Riemann--Hilbert branches: one for $|k|<1$, one for $k>1$, and one for $k<-1$. The branch for $|k|<1$ is positivity preserving, whereas the branches for $|k|>1$ give sign-changing Poisson kernels, providing a disk analogue of Axelsson's half-space example. Finally, we show that these kernels can be realized beyond the $L^2$ class for suitable data, and that their non-uniqueness is intrinsic to the Riemann-Hilbert branch structure rather than to the $L^2$ threshold $|k|=1$.