On certain minimax problems and Pontryagin’s maximum principle

Abstract

This paper deals with minimax problems for nonlinear differential expressions involving a vector-valued function of a scalar variable under rather conventional structure conditions on the cost function. It is proved that an absolutely minimizing (i.e. globally and locally minimizing) function is continuously differentiable. A minimizing function is also continuously differentiable, provided a certain extra condition is satisfied. The variational method of V.G. Boltyanskii, developed within optimal control theory, is adapted and used in the proof. The case of higher order derivatives is also considered.

Existence, Uniqueness and Structure of Second Order Absolute Minimisers

“…Preliminary investigations on the second order 1-dimensional case H(·, u, u , u ) had previously been performed via different methods by Aronsson and Aronsson-Barron in [6,7]. Very recently, arguments inspired by the present paper have been applied in [34] to eigenvalue problems for the ∞-Bilaplacian.…”

“…We recall that the fully nonlinear PDE above has essentially been derived from L p -approximate equations and studied in the paper [32] (see also [6,7]). Note also that in contrast to the system (1.5), (1.6), equation (1.7) does not give uniqueness of strong solutions without additional supplementing conditions.…”

A minimisation problem in L∞with PDE and unilateral constraints

“…We conclude this lengthy introduction with some comments about the general variational context we use herein. Calculus of Variations in L ∞ is a modern subarea of analysis pioneered by Aronsson in the 1960s (see [6]- [9]) who considered variational problems of supremal functionals, rather than integral functional. For a pedagogical introduction we refer e.g.…”

An $L^\infty$ Regularization Strategy to the Inverse Source Identification Problem for Elliptic Equations

“…The field has been initiated in the 1960s by Gunnar Aronsson (see e.g. [3,4,5,6,7]) and is still a very active area of research; for a review of the by-now classical theory involving scalar first order functionals we refer to [21]. To this end, we provide a regularisation strategy inspired by the classical Tykhonov regularisation strategy in L 2 (see e.g.…”