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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 $\\mathrm{F}$. 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