2017 # 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 exceptiona…

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“…[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.…”

confidence: 99%

“…[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.…”

confidence: 99%

“…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.…”

confidence: 88%

“…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.…”

confidence: 99%