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

Abstract

We construct an explicit sign-changing Poisson kernel for a uniformly elliptic divergence form operator in the unit disk. The coefficient matrix is non-symmetric, and its skew-symmetric part has a jump discontinuity across a fixed diameter, with the size of the jump determined by $k\in\mathbb R$. For each $1<p<\infty$, the associated $L^p$ Dirichlet problem undergoes a sharp transition as $k$ crosses a $p$-dependent threshold. In the unique solvability regime, the solution operator preserves positivity and is represented by a nonnegative Poisson kernel. Beyond the threshold, the homogeneous problem admits a nontrivial solution, and for real-valued boundary data, every bounded linear selection from this nonunique solution family is represented by a sign-changing kernel. Thus, even in bounded domains, uniform ellipticity and the maximum principle in the energy class do not prevent Poisson kernels from changing sign. Using a Riemann-Hilbert formulation, we obtain explicit solution formulas and show that the factorization index governs both nonuniqueness and sign change. To our knowledge, this is the first explicit example of this counterintuitive phenomenon in a bounded domain.