Let the linear second-order
elliptic partial differential equation be given in the normal form
with real-valued coefficients that are entire functions on 𝒞2 and whose coefficient
c(x,y) ≤ 0 on the disk D : x2+ y2≤ 1. Let the initial domain of definition of the
real-valued regular solution v = v(x,y) be D. A local Chebyshev approximation
scheme is given by which global information is determined concerning the location of
the singularities of the principal branch of the analytic continuation of v. This follows
from an error analysis of best approximates taken over certain families of regular
solutions whose singularities are in comp(D). The Bergman and Gilbert
Integral Operator Method is utilized in this function-theoretic extension of
the theorems of S. N. Bernstein and E. B. Saff; these theorems classify the
singularities of analytic functions of one complex-variable via the growth in the
error of Chebyshev approximations taken over rational functions of type
(n,ν).