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

Abstract

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.

Vectorial variational problems in L ∞ constrained by the Navier–Stokes equations*

“…The case of finite p > 2 studied herein is also of independent interest, especially for numerical discretisation schemes in L ∞ (see e.g. [41,42]), but in this paper we treat it mostly as an approximation device to solve efficiently the L ∞ problem. The idea of this approach is based on the observation that, for a fixed essentially bounded function on a domain of finite measure, the L p norm tends to the L ∞ norm of the function as p → ∞.…”

Vectorial variational problems in $L^\infty$ constrained by the Navier-Stokes equations

“…The case of finite p > 2 studied herein is also of independent interest, especially for numerical discretisation schemes in L ∞ (see e.g. [42,43]), but in this paper we treat it mostly as an approximation device to solve efficiently the L ∞ problem. The idea of this approach is based on the observation that, for a fixed essentially bounded function on a domain of finite measure, the L p norm tends to the L ∞ norm of the function as p → ∞.…”