Electron. J. Diff. Eqns., Vol. 2007(2007), No. 86, pp. 1-14.

Bernstein approximations of Dirichlet problems for elliptic operators on the plane

Jacek Gulgowski

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 14, 2007.
Math Subject Classifications: 35J25, 41A10.
Key Words: Dirichlet problems; Bernstein polynomials; global bifurcation.

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Jacek Gulgowski
Institute of Mathematics, University of Gdansk
ul. Wita Stwosza 57, 80-952 Gdansk, Poland
email: dzak@math.univ.gda.pl

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