A pointwise characterisation of the PDE system of vectorial calculus of variations in L∞. Issue 4 (August 2020)

• Record Type: Journal Article
• Authors: Ayanbayev, Birzhan; Katzourakis, Nikos
• Is Part Of: Proceedings. Volume 150:Issue 4(2020)
• Page Start: 1653
• Page End: 1669
• Publication Date: 2020-08
• Subjects: ∞-Laplacian, -- generalised solutions, -- calculus of variations in L∞, -- young measures, -- fully non-linear systems; 35D99, -- 35D40, -- 35J47, -- 35J47, -- 35J92, -- 35J70, -- 35J99; Mathematics -- Periodicals; 510
• DOI: 10.1017/prm.2018.89
• Languages: English
• ISSNs: 0308-2105

Abstract

Abstract: Let n, $N \in {\open N}$ with $\Omega \subseteq {\open R}^n$ open. Given ${\rm H} \in C^2(\Omega \times {\open R}^N \times {\open R}^{Nn})$, we consider the functional 1 $${\rm E}_\infty (u, {\rm {\cal O}})\, : = \, \mathop {{\rm ess}\, \sup}\limits_{\rm {\cal O}} {\rm H}(\cdot, u, {\rm D}u), \quad u\in W_{{\rm loc}}^{1, \infty} (\Omega, {\open R}^N), \quad {\rm {\cal O}}{\Subset}\Omega.$$ The associated PDE system which plays the role of Euler–Lagrange equations in $L^\infty $ is 2 $$\left\{\matrix{{\rm H}_{P}(\cdot, u, {\rm D}u)\, {\rm D}\left({\rm H}(\cdot, u, {\rm D} u)\right) = \, 0, \hfill \cr {\rm H}(\cdot, u, {\rm D} u) \, [\![{\rm H}_{P}(\cdot, u, {\rm D} u)]\!]^\bot \left({\rm Div}\left({\rm H}_{P}(\cdot, u, {\rm D} u)\right)- {\rm H}_{\eta}(\cdot, u, {\rm D} u)\right) = 0, \hfill}\right.$$ where $[\![A]\!]^\bot := {\rm Proj}_{R(A)^\bot }$ . Herein we establish that generalised solutions to (2 ) can be characterised as local minimisers of (1 ) for appropriate classes of affine variations of the energy. Generalised solutions to (2 ) are understood as ${\cal D}$ -solutions, a general framework recently introduced by one of the authors.