Existence, uniqueness and characterisation of local minimisers in higher order Calculus of Variations in $\mathrm L^{\infty}$
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arXiv
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| Natura: | Preprint |
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2024
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| author | Katzourakis, Nikos Moser, Roger |
| author_facet | Katzourakis, Nikos Moser, Roger |
| contents | We study variational problems for second order supremal functionals $\mathrm F_\infty(u)= \|F(\cdot,u,\mathrm D u,\mathrm{A}\!:\!\mathrm D^2u)\|_{\mathrm L^{\infty}(Ω)}$, where $F$ satisfies certain natural assumptions, $\mathrm A$ is a positive matrix, and $Ω\Subset \mathbb R^n$. Higher order problems are very novel in the Calculus of Variations in $\mathrm L^{\infty}$, and exhibit a strikingly different behaviour compared to first order problems, for which there exists an established theory, pioneered by Aronsson in 1960s. The aim of this paper is to develop a complete theory for $\mathrm F_\infty$. We prove that, under appropriate conditions, ``localised" minimisers can be characterised as solutions to a nonlinear system of PDEs, which is different from the corresponding Aronsson equation for $\mathrm F_\infty$; the latter is only a necessary, but not a sufficient condition for minimality. We also establish the existence and uniqueness of localised minimisers subject to Dirichlet conditions on $\partial Ω$, and also their partial regularity outside a singular set of codimension one, which may be non-empty even if $n=1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_12625 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence, uniqueness and characterisation of local minimisers in higher order Calculus of Variations in $\mathrm L^{\infty}$ Katzourakis, Nikos Moser, Roger Analysis of PDEs 49K20, 35A15, 35B38, 35D99, 35J94, 49J27 We study variational problems for second order supremal functionals $\mathrm F_\infty(u)= \|F(\cdot,u,\mathrm D u,\mathrm{A}\!:\!\mathrm D^2u)\|_{\mathrm L^{\infty}(Ω)}$, where $F$ satisfies certain natural assumptions, $\mathrm A$ is a positive matrix, and $Ω\Subset \mathbb R^n$. Higher order problems are very novel in the Calculus of Variations in $\mathrm L^{\infty}$, and exhibit a strikingly different behaviour compared to first order problems, for which there exists an established theory, pioneered by Aronsson in 1960s. The aim of this paper is to develop a complete theory for $\mathrm F_\infty$. We prove that, under appropriate conditions, ``localised" minimisers can be characterised as solutions to a nonlinear system of PDEs, which is different from the corresponding Aronsson equation for $\mathrm F_\infty$; the latter is only a necessary, but not a sufficient condition for minimality. We also establish the existence and uniqueness of localised minimisers subject to Dirichlet conditions on $\partial Ω$, and also their partial regularity outside a singular set of codimension one, which may be non-empty even if $n=1$. |
| title | Existence, uniqueness and characterisation of local minimisers in higher order Calculus of Variations in $\mathrm L^{\infty}$ |
| topic | Analysis of PDEs 49K20, 35A15, 35B38, 35D99, 35J94, 49J27 |
| url | https://arxiv.org/abs/2403.12625 |