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

Department of Mathematics and Statistics, University of Reading, Whiteknights, PO Box 220, Reading RG6 6AX, UK.

^** Corresponding author: n.katzourakis@reading.ac.uk

Received: 25 December 2018
Accepted: 29 April 2019

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.