Vol. 91, No. 2, 1980

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Singularities of solutions to linear second order elliptic partial differential equations with analytic coefficients by approximation methods

Peter A. McCoy

Vol. 91 (1980), No. 2, 397–406

Let the linear second-order elliptic partial differential equation be given in the normal form

Δ2v + a(x,y)vx + b(x,y)vv + c(x,y)v = 0, (x,y) ∈ E2

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).

Mathematical Subject Classification 2000
Primary: 35J15
Secondary: 35A20
Received: 30 January 1979
Published: 1 December 1980
Peter A. McCoy