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Department of Mathematics and Statistics, University of Reading, Whiteknights, PO Box 220, ReadingRG6 6AXBerkshire, UK ([email protected]; [email protected])
Nikos Katzourakis
Affiliation:
Department of Mathematics and Statistics, University of Reading, Whiteknights, PO Box 220, ReadingRG6 6AXBerkshire, UK ([email protected]; [email protected])
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
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.
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