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Archive for Rational Mechanics and Analysis

, Volume 231, Issue 3, pp 1615–1634 | Cite as

Existence, Uniqueness and Structure of Second Order Absolute Minimisers

  • Nikos Katzourakis
  • Roger MoserEmail author
Article

Abstract

Let \({\Omega \subseteq \mathbb{R}^n}\) be a bounded open C1,1 set. In this paper we prove the existence of a unique second order absolute minimiser \({u_\infty}\) of the functional
$$\mathrm{E}_\infty (u,\mathcal{O})\, :=\, \|\mathrm{F}(\cdot, \Delta u) \|_{L^\infty( \mathcal{O} )},\,\, \mathcal{O} \subseteq \Omega\,\, \text{measurable},$$
with prescribed boundary conditions for u and \({\mathrm{D}u}\) on \({\partial \Omega}\) and under natural assumptions on F. We also show that \({u_\infty}\) is partially smooth and there exists a harmonic function \({f_\infty \in L^1(\Omega)}\) such that
$${\rm F}(x, \Delta u_\infty(x)) \, =\, e_\infty\,\mathrm{sgn}\big(f_\infty(x)\big)$$
for all \({x \in \{f_\infty \neq 0\}}\) , where \({e_\infty}\) is the infimum of the global energy.

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Notes

Acknowledgements

N.K.would like to thank Craig Evans, Robert Jensen, Jan Kristensen, Juan Manfredi, Giles Shaw and Tristan Pryer for inspiring scientific discussions on the topic of L variational problems.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of Reading, WhiteknightsReadingUK
  2. 2.Department of Mathematical SciencesUniversity of Bath, Claverton DownBathUK

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