Bernstein approximations of Dirichlet problems for elliptic operators on the plane

Abstract

We study the finitely dimensional approximations of the elliptic problem (Lu)(x, y) + φ(λ, (x, y), u(x, y)) = 0 for (x, y) ∈ Ω u(x, y) = 0 for (x, y) ∈ ∂Ω, defined for a smooth bounded domain Ω on a plane. The approximations are derived from Bernstein polynomials on a triangle or on a rectangle containing Ω. We deal with approximations of global bifurcation branches of nontrivial solutions as well as certain existence facts.