Laplace problems on planar domains can be solved by means of least-squares expansions associated with polynomial or rational approximations. Here it is shown that, even in the context of an analytic domain with analytic boundary data, the difference in convergence rates may be huge when the domain is nonconvex. Our proofs combine the theory of the Schwarz function for analytic continuation, potential theory for polynomial and rational approximation rates, and the theory of crowding of conformal maps.

翻译：平面域上的拉普拉斯问题可通过多项式或有理逼近的最小二乘展开求解。本文证明，即使对于具有解析边界数据的解析区域，当区域非凸时，两种逼近方法的收敛速率可能存在巨大差异。我们的证明综合运用了解析延拓的施瓦兹函数理论、多项式与有理逼近速率的位势理论，以及共形映射的聚集理论。