Absolutely minimising generalised solutions to the equations of vectorial calculus of variations in $$L^\infty $$ L ∞

Abstract

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.

Explicit $$\infty$$ ∞ -harmonic functions in high dimensions

“…[9]). Since then, the field has been developed enormously by N. Katzourakis in the series of papers ( [11][12][13][14][15][16][17][18][19]) and also in collaboration with the author, Abugirda, Croce, Manfredi, Moser, Parini, Pisante and Pryer ( [1,2,6,[20][21][22][23][24][25]). A standard difficulty of (1.1) is that it is nondivergence form equation and since in general smooth solutions do not exist, the definition of generalised solutions is an issue.…”

On the numerical approximation of $$\infty $$ ∞ -harmonic mappings

“…This is true in the scalar case N = 1 in the sense of viscosity solutions of Crandall-Ishii-Lions [14,36]. In the vectorial case when n = 1, it is true in the sense of D-solutions of [27]. We conjecture this to also be true in the case of (1.1) when both n, N > 1, but this is not a consequence of the current results of [26] since the method of the existence proof was based on an ad-hoc method (an analytic counterpart of Gromov's "Convex Integration" for a differential inclusion) rather than on p-harmonic approximations.…”

“…Since we do not utilise it in an essential manner in this paper, we refrain from giving all the details which can be found in [26] and [27][28][29][30]. The idea applies to general fully nonlinear systems of any order and allows for merely measurable solutions.…”

“…Up to the early 2010s, all considerations were restricted to the scalar case. The general vectorial case of (1.1) as well as the study of the associated PDE systems arising from more general first order functionals (1.5) has been initiated in a series of recent papers [20][21][22][23][24][25][26][27][28]. Besides the intrinsic mathematical interest of the field, L ∞ functionals can be applied in a variety of scenarios.…”

“…The latter may not exist classically and we define it "weakly" in a duality-free manner by considering the limits of difference quotients as Young measures. In this setting, existence of a solution to the Dirichlet problem (1.1) and for the more general class of equations arising from (1.5) for n = 1 has been proven [27]. Another difficulty is that solutions to (1.1) are not in general unique [24].…”