# Second order L∞ variational problems and the ∞-polylapacian

(2020) Second order L∞ variational problems and the ∞-polylapacian. Advances in Calculus of Variations, 13 (2). pp. 115-140. ISSN 1864-8266

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To link to this item DOI: 10.1515/acv-2016-0052

## Abstract/Summary

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 $\mathrm{H}\in C^1(\mathbb{R}^{n\times n}_s)$, for the functional $\label{1} \ \ \ \ \mathrm{E}_\infty(u,\mathcal{O})\, =\, \big\| \mathrm{H}\big(\mathrm{D}^2 u\big) \big\|_{L^\infty(\mathcal{O})}, \ \ \ u\in W^{2,\infty}(\Omega),\ \mathcal{O}\subseteq \Omega, \tag{1}$ the associated equation is the fully nonlinear $3$rd order PDE $\label{2} \A^2_\infty u\, :=\,\big(\mathrm{H}_X\big(\mathrm{D}^2u\big)\big)^{\otimes 3}:\big(\mathrm{D}^3u\big)^{\otimes 2}\, =\,0. \tag{2}$ 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.

Item Type: Article Yes Faculty of Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics 74829 De Gruyter