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.