Existence, Uniqueness and Structure of Second Order absolute minimisers

Abstract

Let $Ω\subseteq \mathbb{R}^n$ 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 $\partial Ω$ and under natural assumptions on $\mathrm{F}$. We also show that $u_\infty$ is partially smooth and there exists a harmonic function $f_\infty \in L^1(Ω)$ such that \[ \mathrm{F}(x, Δu_\infty(x)) \, =\, e_\infty\, \mathrm{sgn}\big(f_\infty(x)\big) \] for all $x \in \{f_\infty \neq 0\}$, where $e_\infty$ is the infimum of the global energy.

17 pages; Journal: Archives for Rational Mechanics and Analysis