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

2026/06/30 by Seick Kim
#math.AP

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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 controlled by a real parameter k. Using a first-order Dirac formulation, we obtain explicit solution formulas, L² non-tangential maximal estimates, and almost everywhere non-tangential convergence to the prescribed boundary data. We show that the associated L² boundary equation undergoes a sharp transition at |k|=1, giving rise to three natural L² 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 and k<-1 yield 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 L² class for suitable boundary data and that the resulting non-uniqueness arises from the Riemann-Hilbert branch structure rather than from the L² threshold |k|=1.

Citations

• Square function/non-tangential maximal function estimates and the\n dirichlet problem for non-symmetric elliptic operators
• Weighted maximal regularity estimates and solvability of non-smooth\n elliptic systems I
• Solvability of elliptic systems with square integrable boundary data
• Non unique solutions to boundary value problems for non symmetric divergence form equations
• Functional calculus of Dirac operators and complex perturbations of\n Neumann and Dirichlet problems
• A New Approach to Absolute Continuity of Elliptic Measure, with Applications to Non-symmetric Equations
• The neumann problem for elliptic equations with non-smooth coefficients
• Weighted norm inequalities for the conjugate function and Hilbert transform

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