Strong convergence of numerical solutions to degenerate variational problems

Abstract:

Numerical approximations of strongly degenerate variational problems of the form $J(u) = \smallint _0^1F(u’ ) + {(u - f)^2}$ are considered, where F is assumed convex but may have intervals where $F” = 0$. It is shown that, in spite of the degeneracy, natural numerical approximations still converge in ${W^{1,p}}$. Rates in weaker norms and the connection with nonconvex variational problems are also considered.