Dual and inverse formulations of constrained extremum problems

Abstract

In this paper we consider constrained extremum problems of th form Pp:infu∈t−1(p)f(u) where f and t are continuously differentiable functionals on a reflexive Banach space V and where t-1(p) denotes the level set of the functional t with value p ϵ R. Related to problems Pp we investigate inverse extremum problems, which are extremum problems for the functional t on level sets of the functional f. Under conditions that guarantee the existence of solutions of Pp, let h(p) denote the value of f at such a solution. If h is a (locally) convex function at some p̄ ϵ R, we show that it is possible to define a dual problem of Pp̄. This dual problem is a saddle-point formulation for the functional V × RЭ(u,μ)→f(u)−μ[t(u)−p̄]: for some extreme value μ̄ (which is the Lagrange multiplier of a solution of Pp̄) the solutions of Pp̄ are precisely the (local) minimal points of the functional f − μ̄t on V. It is shown how these results can be used to describe solution branches of nonlinear eigenvalue problems (of semilinear elliptic type) with a global parameter, such as p ϵ R, instead of with the eigenvalue as a parameter.