# A minimisation problem in ${\mathrm{L}}^\infty$ with PDE and unilateral constraints

## TL;DR

This paper addresses a PDE-constrained optimization problem in ${
m L}^
Infty$ involving boundary misfit measurement and unilateral constraints, employing approximation and variational inequalities, motivated by biomedical imaging applications.

## Contribution

It introduces a novel ${
m L}^
Infty$ minimization framework with PDE and unilateral constraints, using approximation and Kuhn-Tucker theory, applied to biomedical imaging.

## Key findings

- Derived variational inequalities in ${
m L}^p$ and ${
m L}^
Infty$
- Established approximation approach via ${
m L}^p$ problems as $p o
Infty$
- Motivated by and applicable to Fluorescent Optical Tomography

## 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 ${\mathrm{L}}^\infty$, where except for the PDE constraint there is also a unilateral constraint on the parameter. We utilise approximation by PDE-constrained minimisation problems in ${\mathrm{L}}^p$ as $p\to\infty$ and the generalised Kuhn-Tucker theory to derive the relevant variational inequalities in ${\mathrm{L}}^p$ and ${\mathrm{L}}^\infty$. These results are motivated by the mathematical modelling of the novel bio-medical imaging method of Fluorescent Optical Tomography.