Existence, Uniqueness and Structure of Second Order absolute minimisers

Abstract

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

17 pages; Journal: Archives for Rational Mechanics and Analysis

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