Volume 26, 2020
|Number of page(s)||27|
|Published online||08 September 2020|
A minimisation problem in L∞ with PDE and unilateral constraints*
Department of Mathematics and Statistics, University of Reading, Whiteknights,
PO Box 220,
RG6 6AX, UK.
** Corresponding author: email@example.com
Accepted: 29 April 2019
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 Lp as p →∞ and the generalised Kuhn-Tucker theory to derive the relevant variational inequalities in Lp and L∞. These results are motivated by the mathematical modelling of the novel bio-medical imaging method of Fluorescent Optical Tomography.
Mathematics Subject Classification: 35Q93 / 49K20 / 49J40
Key words: Absolute minimisers / calculus of variations in L∞ / PDE-constrained optimisation / generalised Kuhn–Tucker theory / Lagrange multipliers / fluorescent optical tomography / Robin boundary conditions
© EDP Sciences, SMAI 2020
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