A pointwise characterisation of the PDE system of vectorial calculus of variations in L ∞

Advances in Calculus of Variations

Pryer

2018

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In this paper we initiate the study of 2nd order variational problems in L ∞ , seeking to minimise the L ∞ norm of a function of the hessian. We also derive and study the respective PDE arising as the analogue of the Euler-Lagrange equation. Given H ∈ C 1 (R n×n s ), for the functionalthe associated equation is the fully nonlinear 3rd order PDESpecial cases arise when H is the Euclidean length of either the full hessian or of the Laplacian, leading to the ∞-Polylaplacian and the ∞-Bilaplacian respectively. We establish several results for (1) and (2), including existence of minimisers, of absolute minimisers and of "critical point" generalised solutions, proving also variational characterisations and uniqueness. We also construct explicit generalised solutions and perform numerical experiments.

Section: Existence Of 2nd Order Minimisers and Absolute Minimisersmentioning

confidence: 99%

Second-order L ^∞ variational problems and the ∞-polylaplacian

Advances in Calculus of Variations

Pryer

2018

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“…This is connected to that fact that, when N ≥ 2, the associated Euler-Lagrange ∞-Laplace system admits smooth non-minimising solutions which are characterised by two distinct variational concepts, see e.g. [3,30,36].…”

Section: Definition 1 (Absolute Minimisers)mentioning

confidence: 99%

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

Nonlinear Differ. Equ. Appl.

Pryer

2020

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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.

“…Diffuse derivatives can be seen as measure-theoretic disintegrations whose barycentres are the distributional derivatives (see [29]). For further results relevant to D-solutions and their applications, see [30]- [32], [34,10,35,13], [36]- [38].…”

Section: Nikos Katzourakismentioning

confidence: 99%

Weak vs. 𝒟-solutions to linear hyperbolic first-order systems with constant coefficients

J. Hyper. Differential Equations

2018

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We establish a consistency result by comparing two independent notions of generalised solutions to a large class of linear hyperbolic first order PDE systems with constant coefficients, showing that they eventually coincide. The first is the usual notion of weak solutions defined via duality. The second is the new notion of D-solutions introduced in the recent paper [29], which arose in connection to vectorial Calculus of Variations in L ∞ and fully nonlinear elliptic systems. This new approach is a duality-free alternative to distributions and is based on the probabilistic representation of limits of difference quotients.