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Existence, Uniqueness and Structure of Second Order Absolute Minimisers

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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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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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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Reading, Whiteknights, PO Box 220, Reading, RG6 6AX, UK

    Nikos Katzourakis

  2. Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK

    Roger Moser

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  1. Nikos Katzourakis
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  2. Roger Moser
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Correspondence to Roger Moser.

Additional information

Communicated by C. De Lellis

N.K. has been partially financially supported by the EPSRC Grant EP/N017412/1.

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Katzourakis, N., Moser, R. Existence, Uniqueness and Structure of Second Order Absolute Minimisers. Arch Rational Mech Anal 231, 1615–1634 (2019). https://doi.org/10.1007/s00205-018-1305-6

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  • DOI: https://doi.org/10.1007/s00205-018-1305-6

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