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.

We introduce a new theory of generalised solutions which applies to fully nonlinear PDE systems of any order and allows for merely measurable maps as solutions. This approach bypasses the standard problems arising by the application of Distributions to PDEs and is not based on either integration by parts or on the maximum principle. Instead, our starting point builds on the probabilistic representation of derivatives via limits of difference quotients in the Young measures over a toric compactification of the space of jets. After developing some basic theory, as a first application we consider the Dirichlet problem and we prove existence-uniqueness-partial regularity of solutions to fully nonlinear degenerate elliptic 2nd order systems and also existence of solutions to the ∞-Laplace system of vectorial Calculus of Variations in L ∞ .

Let Ω ⊆ R n be a bounded open C 1,1 set. In this paper we prove the existence of a unique second order absolute minimiser u∞ of the functionalwith prescribed boundary conditions for u and Du on ∂Ω and under natural assumptions on F. We also show that u∞ is partially smooth and there exists a harmonic function f∞ ∈ L 1 (Ω) such that F(x, ∆u∞(x)) = e∞ sgn f∞(x)for all x ∈ {f∞ = 0}, where e∞ is the infimum of the global energy.

Consider the supremal functional (1) E∞(u, A) := L (·, u, Du) L ∞ (A) , A ⊆ Ω,Under certain assumptions on L , we prove for any given boundary data the existence of a map which is: i) a vectorial Absolute Minimiser of (1) in the sense of Aronsson, ii) a generalised solution to the ODE system associated to (1) as the analogue of the Euler-Lagrange equations, iii) a limit of minimisers of the respective L p functionals as p → ∞ for any q ≥ 1 in the strong W 1,q topology & iv) partially C 2 on Ω off an exceptional compact nowhere dense set. Our method is based on L p approximations and stable a priori partial regularity estimates. For item ii) we utilise the recently proposed by the author notion of D-solutions in order to characterise the limit as a generalised solution. Our results are motivated from and apply to Data Assimilation in Meteorology.

Let H ∈ C 2 (R N ×n ), H ≥ 0. The PDE system (1) A∞u := H P ⊗ H P + H[H P ] ⊥ H P P (Du) : D 2 u = 0 arises as the "Euler-Lagrange PDE" of vectorial variational problems for the functional E∞(u, Ω) = H(Du) L ∞ (Ω) defined on maps u : Ω ⊆ R n −→ R N . (1) first appeared in the author's recent work [K3]. The scalar case though has a long history initiated by Aronsson in [A1]. Herein we study the solutions of (1) with emphasis on the case of n = 2 ≤ N with H the Euclidean norm on R N ×n , which we call the "∞-Laplacian". By establishing a rigidity theorem for rank-one maps of independent interest, we analyse a phenomenon of separation of the solutions to phases with qualitatively different behaviour. As a corollary, we extend to N ≥ 2 the Aronsson-Evans-Yu theorem regarding non-existence of zeros of |Du| and prove a Maximum Principle. We further characterise all H for which (1) is elliptic and also study the initial value problem for the ODE system arising for n = 1 but with H(·, u, u ) depending on all the arguments.

The ∞‐Bilaplacian is a third‐order fully nonlinear PDE given by
Δ∞2u≔()Δu3||normalD()Δu2=0.
In this work, we build a numerical method aimed at quantifying the nature of solutions to this problem, which we call ∞‐biharmonic functions. For fixed p we design a mixed finite element scheme for the prelimiting equation, the p‐Bilaplacian
Δp2u≔Δ||Δup−2Δu=0.
We prove convergence of the numerical solution to the weak solution of Δp2u=0 and show that we are able to pass to the limit p → ∞. We perform various tests aimed at understanding the nature of solutions of Δ∞2u and we prove convergence of our discretization to an appropriate weak solution concept of this problem that of D‐solutions.

Abstract. A map u : Ω ⊆ R n −→ R N , is said to be ∞-harmonic if it satisfies(1)The system (1) is the model of vector-valued Calculus of Variations in L ∞ and arises as the "Euler-Lagrange" equation in relation to the supremal functional(2)In this work we provide numerical approximations of solutions to the Dirichlet problem when n = 2 and in the vector valued case of N = 2, 3 for certain carefully selected boundary data on the unit square. Our experiments demonstrate interesting and unexpected phenomena occurring in the vector valued case and provide insights on the structure of general solutions and the natural separation to phases they present.

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