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Second order L∞ variational problems and the ∞-polylapacian

Katzourakis, N. and Pryer, T. (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
Refereed:Yes
Divisions:Faculty of Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
ID Code:74829
Publisher:De Gruyter

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