Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 86, pp. 1-14.

Title: Bernstein approximations of Dirichlet problems for
       elliptic operators on the plane

Author: Jacek Gulgowski (Univ. of Gdansk,  Poland)

Abstract:
 We study the finitely dimensional approximations of the elliptic problem
 $$\displaylines{
 (Lu)(x,y) + \varphi(\lambda,(x,y),u(x,y) ) = 0 \quad
 \hbox{for } (x,y)\in\Omega\cr
 u(x,y) = 0 \quad \hbox{for } (x,y)\in\partial\Omega,
 }$$
 defined for a smooth bounded domain $\Omega$ on a plane.
 The approximations are derived from Bernstein polynomials on a triangle or
 on a rectangle containing $\Omega$. We deal with approximations of
 global bifurcation branches of nontrivial solutions
 as well as certain existence facts.
  
Submitted January 2, 2007. Published June 15, 2007.
Math Subject Classifications: 35J25, 41A10.
Key Words: Dirichlet problems; Bernstein polynomials; global bifurcation.