Second Order $L^{\infty }$ Variational Problems and the $\infty$-Polylaplacian

Abstract

In this paper we initiate the study of $2$nd order variational problems in $L^{\infty }$, seeking to minimise the $L^{\infty }$ norm of a function of the hessian. We also derive and study the respective PDE arising as the analogue of the Euler-Lagrange equation. Given $\mathrm{H}\in C^1({\mathbb{R}}_s^{n\times n})$, for the functional \[ \label{1} \mathrm{E}_\infty(u,\mathcal{O})\, =\, \big\| \mathrm{H}\big(\mathrm{D}^2 u\big) \big\|_{L^\infty(\mathcal{O})}, \ \ \ u\in W^{2,\infty}(Ω),\ \mathcal{O}\subseteq Ω, \tag{1} \] the associated equation is the fully nonlinear 3rd order PDE \[ \label{2} \mathrm{A}^2_\infty u\, :=\,\big(\mathrm{H}_X\big(\mathrm{D}^2u\big)\big)^{\otimes 3}:\big(\mathrm{D}^3u\big)^{\otimes 2}\, =\,0. \tag{2} \] Special cases arise when $\mathrm{H}$ is the Euclidean length of either the full hessian or of the Laplacian, leading to the $\infty$-Polylaplacian and the $\infty$-Bilaplacian respectively. We establish several results for \eqref{1} and \eqref{2}, including existence of minimisers, of absolute minimisers and of "critical point" generalised solutions, proving also variational characterisations and uniqueness. We also construct explicit generalised solutions and perform numerical experiments.

Journal: Advances in Calculus of Variations, 31 pages, 13 figures

References in corpus (6)

• Absolutely Minimising Generalised Solutions to the Equations of Vectorial Calculus of Variations in $L^{\infty }$
• A New Characterisation of $\infty$-Harmonic and $p$-Harmonic Maps via Affine Variations in $L^{\infty }$
• Mollification of $\mathcal{D}$-solutions to Fully Nonlinear PDE Systems
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Cited by in corpus (7)

• Absolutely Minimising Generalised Solutions to the Equations of Vectorial Calculus of Variations in $L^{\infty }$
• A New Characterisation of $\infty$-Harmonic and $p$-Harmonic Maps via Affine Variations in $L^{\infty }$
• A Pointwise Characterisation of the PDE System of Vectorial Calculus of Variations in $L^{\infty }$
• Solutions of Vectorial Hamilton-Jacobi Equations are Rank-One Absolute Minimisers in $L^{\infty }$
• On the numerical approximation of $p$-Biharmonic and $\infty$-Biharmonic functions
• Explicit $\infty$-harmonic functions in high dimensions
• The Eigenvalue Problem for the $\infty$-Bilaplacian