On Approximation of Linear Second Order Elliptic Partial Differential Equations with Analytic Coefficients

Abstract

The linear second-order elliptic differential equation with real-valued coefficients that are entire functions on $\Im^2$ and whose coefficient $c(x, y) \leq 0$ on the disk $D : x^2+y^2\leq1$ is given by $\Delta^2 v+a(x,y)v_x + b(x,y)v_y+c(x,y)v=0, (x,y)\in E^2$. The ideas of Bernstein and Saff have been applied by McCoy [9, 10] to study the singularities of certain second-order elliptic equations with singular coefficients. These results contain calculations of order and type of entire function potentials in terms of best polynomial approximation errors. Here some inequalities concerning order and type for the given equation have been obtained.