A sufficient condition for the lower semicontinuity of nonlocal supremal functionals in the vectorial case

In this note we present a sufficient condition ensuring lower semicontinuity for nonlocal supremal functionals of the type $$W^{1,\infty}(\Omega;\mathbb R^d)\ni u \mapsto \sup{\rm ess}_{(x,y)\in \Omega} W(x,y, \nabla u(x),\nabla u(y)),$$ where $\Omega$ is a bounded open subset of $\mathbb R^N$ and $W:\Omega \times \Omega \times \mathbb R^{d \times N}\times \mathbb R^{d \times N} \to \mathbb R$.Comment: to appear in European Journal of Mathematic

Vectorial variational principles in L∞ and their characterisation through PDE systems

We discuss two distinct minimality principles for general supremal first order functionals for maps and characterise them through solvability of associated second order PDE systems. Specifically, we consider Aronsson’s standard notion of absolute minimisers and the concept of ∞ -minimal maps introduced more recently by the second author. We prove that C1 absolute minimisers characterise a divergence system with parameters probability measures and that C2∞ -minimal maps characterise Aronsson’s PDE system. Since in the scalar case these different variational concepts coincide, it follows that the non-divergence Aronsson’s equation has an equivalent divergence counterpart

Vectorial variational problems in L ∞ constrained by the Navier–Stokes equations * * EC has been financially supported through the UK EPSRC scholarship GS19-055. BM has been partially financially supported through the Croatian Science Foundation project IP-2019-04-1140.

We study a minimisation problem in L p and L ∞ for certain cost functionals, where the class of admissible mappings is constrained by the Navier–Stokes equations. Problems of this type are motivated by variational data assimilation for atmospheric flows arising in weather forecasting. Herein we establish the existence of PDE-constrained minimisers for all p, and also that L p minimisers converge to L ∞ minimisers as p → ∞. We further show that L p minimisers solve an Euler–Lagrange system. Finally, all special L ∞ minimisers constructed via approximation by L p minimisers are shown to solve a divergence PDE system involving measure coefficients, which is a divergence-form counterpart of the corresponding non-divergence Aronsson–Euler system

A minimisation problem in L∞ with PDE and unilateral constraints

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

On the inverse source identification problem in L ∞ for fully nonlinear elliptic PDE

Abstract: In this paper we generalise the results proved in N. Katzourakis (SIAM J. Math. Anal. 51, 1349–1370, 2019) by studying the ill-posed problem of identifying the source of a fully nonlinear elliptic equation. We assume Dirichlet data and some partial noisy information for the solution on a compact set through a fully nonlinear observation operator. We deal with the highly nonlinear nonconvex nature of the problem and the lack of weak continuity by introducing a two-parameter Tykhonov regularisation with a higher order L2 “viscosity term” for the L∞ minimisation problem which allows to approximate by weakly lower semicontinuous cost functionals

An L∞ regularisation strategy to the inverse source identification problem for elliptic equations

In this paper we utilise new methods of Calculus of Variations in L ∞ to provide a regularisation strategy to the ill-posed inverse problem of identifying the source of a non-homogeneous linear elliptic equation, satisfying Dirichlet data on a domain. One of the advantages over the classical Tykhonov regularisation in L 2 is that the approximated solution of the PDE is uniformly close to the noisy measurements taken on a compact subset of the domain