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 with non-symmetric coefficients having a jump discontinuity across a diameter. The skew-symmetric part is controlled by a real parameter $k$, while ellipticity is preserved for all $k$. Using a first-order Dirac formulation, we obtain an explicit Poisson-type representation of solutions. The formula shows that the boundary equation changes its character at $|k|=1$. For $|k|<1$, it gives the natural positivity-preserving solution. For $|k|>1$, another branch of solutions appears, and this branch accounts for the sign-changing Poisson kernel previously constructed by Axelsson in the half-space model. The result gives a bounded-domain explanation of this phenomenon and identifies the mechanism behind the change of behavior as the parameter $k$ varies.