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

Content Provider	Springer Nature Link
Author	Katzourakis, Nikos
Copyright Year	2017
Abstract	Consider the supremal functional 1 $$\begin{aligned} E_\infty (u,A) := \Vert \mathscr {L}(\cdot ,u,\mathrm {D}u)\Vert _{L^\infty (A)},\quad A\subseteq \Omega , \end{aligned}$$ applied to $$W^{1,\infty }$$ maps $$u:\Omega \subseteq \mathbb {R}\longrightarrow \mathbb {R}^N$$ , $$N\ge 1$$ . Under certain assumptions on $$\mathscr {L}$$ , 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 $$L^p$$ L p functionals as $$p\rightarrow \infty $$ p → ∞ for any $$q\ge 1$$ q ≥ 1 in the strong $$W^{1,q}$$ W 1 , q topology and (iv) partially $$C^2$$ C 2 on $$\Omega $$ 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 $$\mathcal {D}$$ -solutions in order to characterise the limit as a generalised solution. Our results are motivated from and apply to Data Assimilation in Meteorology.
Ending Page	25
Page Count	25
Starting Page	1
File Format	PDF
ISSN	09442669
e-ISSN	14320835
Journal	Calculus of Variations and Partial Differential Equations
Issue Number	1
Volume Number	56
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2017-01-16
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Systems Theory, Control Calculus of Variations and Optimal Control; Optimization Second-order elliptic systems Analysis Theoretical, Mathematical and Computational Physics Quasilinear elliptic equations Rigidity results
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis

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