A minimisation problem in L^∞with PDE and unilateral constraints

Abstract

We study the minimisation of a cost functional which measures the misfit on the boundary of a domain between a component of the solution to a certain parametric elliptic PDE system and a prediction of the values of this solution. We pose this problem as a PDE-constrained minimisation problem for a supremal cost functional in L ∞ , where except for the PDE constraint there is also a unilateral constraint on the parameter. We utilise approximation by PDE-constrained minimisation problems in L p as p → ∞ and the generalised Kuhn-Tucker theory to derive the relevant variational inequalities in L p and L ∞ . These results are motivated by the mathematical modelling of the novel bio-medical imaging method of Fluorescent Optical Tomography. 1 a 0 , K, L, M ∈ L ∞ (Ω; R 2×2 ), k 1 , ≥ a 0 , l 1 ≥ a 0 .

Generalized second order vectorial ∞-eigenvalue problems

“…Higher order problems and problems involving constraints present additional difficulties and have been studied even more sparsely, see e.g. [3,4,9,10,[20][21][22][23][24]26]. In fact, this paper is an extension of [23] to the second order case, and generalizes part of the results corresponding to the existence of minimizers and the satisfaction of PDEs from [25].…”

Vectorial variational problems in L ∞ constrained by the Navier–Stokes equations*

“…In this paper, to overcome the difficulties described above, we follow the methodology of the relatively new field of calculus of variations in L ∞ (see e.g. [22,35] for a general introduction to the scalar-valued theory), and in particular the ideas from [37][38][39][40] involving higher order and vectorial problems, as well as problems involving PDE-constraints, which have only recently started being investigated. To this end, we follow the approach of solving the desired L ∞ variational problem by solving respective approximating L p variational problems for all p, and obtain appropriate compactness estimates which allow to pass to the limit as p → ∞.…”

“…[22,35]). However, vectorial and higher L ∞ variational problems involving constraints, have only recently began being explored (see [38,39], but also the relevant earlier contributions [3,6,10]). For several interesting developments on L ∞ variational problems we refer the interested reader to [9,11,14,15,20,26,30,44,[47][48][49].…”

“…In this paper, motivated by recent progress on higher order and on constrained L ∞ variational problems made in [40] by the author jointly with Moser and by the author in [38,39] (inspired by earlier contributions by Moser and Schwetlick deployed in a geometric setting in [46]), we follow a slightly different approach which does not lead an Aronsson-Euler type system; instead, it leads to a divergence structure PDE system. However, there is a toll to be paid, as the divergence PDEs arising as necessary conditions involve measures as auxiliary parameters whose determination becomes part of the problem.…”