On the lower semicontinuity and approximation of $${L^{\infty}}$$ L ∞ -functionals

2020

ESAIM: COCV

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 .

“…Interesting results regarding L ∞ variational problems can be found e.g. in [10,14,15,16,17,18,19,28,43,46,47,48].…”

Section: )mentioning

confidence: 99%

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

2020

ESAIM: COCV

“…Nonetheless, since its inception it has attracted the interest of many mathematicians due to both the theoretical importance (see e.g. [2,10,11,16,22,28,38] and the expository texts [7,15,31]) as well as due to the relevance to various and diverse applications from electrical breakdown [26] to image processing [21] to polycrystals [13] and from conformal mappings [14,29] to game theory [9,37].…”

Section: Introductionmentioning

confidence: 99%

On the numerical approximation of vectorial absolute minimisers in $$L^\infty $$

Nonlinear Differ. Equ. Appl.

Pryer

2020

Let $$\Omega $$ Ω be an open set. We consider the supremal functional $$\begin{aligned} \text {E}_\infty (u,{\mathcal {O}})\, {:}{=}\, \Vert \text {D}u \Vert _{L^\infty ( {\mathcal {O}} )}, \ \ \ {\mathcal {O}} \subseteq \Omega \text { open}, \end{aligned}$$ E ∞ ( u , O ) : = ‖ D u ‖ L ∞ ( O ) , O ⊆ Ω open , applied to locally Lipschitz mappings $$u : \mathbb {R}^n \supseteq \Omega \longrightarrow \mathbb {R}^N$$ u : R n ⊇ Ω ⟶ R N , where $$n,N\in \mathbb {N}$$ n , N ∈ N . This is the model functional of Calculus of Variations in $$L^\infty $$ L ∞ . The area is developing rapidly, but the vectorial case of $$N\ge 2$$ N ≥ 2 is still poorly understood. Due to the non-local nature of (1), usual minimisers are not truly optimal. The concept of so-called absolute minimisers is the primary contender in the direction of variational concepts. However, these cannot be obtained by direct minimisation and the question of their existence under prescribed boundary data is open when $$n,N\ge 2$$ n , N ≥ 2 . We present numerical experiments aimed at understanding the behaviour of minimisers through a new technique involving p-concentration measures.

“…Additionally, in the papers [1,35] a concept of special affine variations was considered which also characterises the Aronsson system, in fact in the generality of merely locally Lipschitz D-solutions. Finally, in the paper [8] new concepts of absolute minimisers for constrained minimisation problems have been proposed, whilst results relevant to variational principles in L ∞ and applications appear in [15,16,18,26,41,42].…”

Section: Introductionmentioning

confidence: 99%

Vectorial Variational Principles in $$L^\infty $$ and Their Characterisation Through PDE Systems

Ayanbayev

2019

Appl Math Optim

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 C 1 absolute minimisers characterise a divergence system with parameters probability measures and that C 2 ∞-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.