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A pointwise characterisation of the PDE system of vectorial calculus of variations in L∞
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 150 / Issue 4 / August 2020
- Published online by Cambridge University Press:
- 01 February 2019, pp. 1653-1669
- Print publication:
- August 2020
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- Article
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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 functional1The associated PDE system which plays the role of Euler–Lagrange equations in
$${\rm E}_\infty (u,{\rm {\cal O}})\, : = \, \mathop {{\rm ess}\,\sup}\limits_{\rm {\cal O}} {\rm H}(\cdot, u,{\rm D}u),\quad u\in W_{{\rm loc}}^{1,\infty} (\Omega, {\open R}^N),\quad {\rm {\cal O}}{\Subset}\Omega.$$
$L^\infty $ is2where
$$\left\{\matrix{{\rm H}_{P}(\cdot, u, {\rm D}u)\, {\rm D}\left({\rm H}(\cdot, u, {\rm D} u)\right) = \, 0, \hfill \cr {\rm H}(\cdot, u, {\rm D} u) \, [\![{\rm H}_{P}(\cdot, u, {\rm D} u)]\!]^\bot \left({\rm Div}\left({\rm H}_{P}(\cdot, u, {\rm D} u)\right)- {\rm H}_{\eta}(\cdot, u, {\rm D} u)\right) = 0,\hfill}\right.$$
$[\![A]\!]^\bot := {\rm Proj}_{R(A)^\bot }$. 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.
Beltrami Equation with Coefficient in Sobolev and Besov Spaces
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- Journal:
- Canadian Journal of Mathematics / Volume 65 / Issue 6 / 01 December 2013
- Published online by Cambridge University Press:
- 20 November 2018, pp. 1217-1235
- Print publication:
- 01 December 2013
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- Article
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Our goal in this work is to present some function spaces on the complex plane
$\mathbb{C},\,X(\mathbb{C})$ , for which the quasiregular solutions of the Beltrami equation, 
$\bar{\partial }f(z)\,=\,\mu (z)\partial f(z)$ , have first derivatives locally in 
$X(\mathbb{C})$ , provided that the Beltrami coefficient 
$\mu $ belongs to 
$X(\mathbb{C})$ .
