For elliptic boundary value problems of the form −ΔU+F(x, U, Ux)=0 on Ω,BU=0 on ϖΩ, with a nonlinearityFgrowing at most quadratically with respect to the gradientUxand with a mixed-type linear boundary opeatorB, a numerical method is presented which can be used to prove the existence of a solution within a “close”H1,4(Ω)-neighborhood of some approximate solution ω∈H2(Ω) satisfying the boundary condition, provided that the defect-norm ∥−Δω+F(·, ω, ωx)∥2is sufficiently small and, moreover, the linearization of the given problem at ω leads to an invertible operatorL. The main tools are explicit Sobolev imbeddings and eigenvalue bounds forLor forL*L. All kinds of monotonicity or inverse-positivi
展开▼
