On second-order $ L^\infty $ variational problems with lower-order terms

Ben Dutton; Nikos Katzourakis

doi:10.3934/dcds.2025154

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2026, Volume 49: 1-21. Doi: 10.3934/dcds.2025154

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On second-order $ L^\infty $ variational problems with lower-order terms

Ben Dutton^1, , and
Nikos Katzourakis^2,

1.
Department of Mathematical Sciences, University of Bath, Bath, United Kingdom
2.
Department of Mathematics and Statistics, University of Reading, Reading, United Kingdom

^*Corresponding author: Ben Dutton
^*Corresponding author: Ben Dutton

Received: May 2025

Revised: September 2025

Early access: October 10, 2025

Published online: October 10, 2025

B.D. has been financially supported by an Undergraduate Research Bursary URB-2023-95 from the London Mathematical Society and a scholarship from the EPSRC CDT in Statistical Applied Mathematics at Bath (SAMBa), under the project EP/Y034716/1.
N.K. has been partially financially supported through the EPSRC grant EP/X017206/1.

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Abstract

In this paper we study $ 2 $nd-order $ L^\infty $-variational problems by seeking to minimise a supremal functional involving the Hessian of admissible functions as well as their lower-order terms, considering for fixed $ \Omega\subseteq\mathbb R^n $ open, and $ {\mathrm{H}} : \Omega\times\big(\mathbb R \times\mathbb R^n \times \mathbb R^{n^{\otimes2}}_s \big) \to \mathbb R $, the functional

$ \ \ \ \ \ {\mathrm{E}}_\infty(u,\mathcal{O}) : = \underset{\mathcal{O}}{\mathrm{ess}\sup} \quad \mathrm H (\cdot,u,\mathrm D u,\mathrm D^2u ) , \ \ u\in \mathrm{W}^{2,\infty}(\Omega), \ \mathcal{O} \subseteq \Omega \text{ measurable }. $

Specifically, we establish the existence of minimisers subject to (first-order) Dirichlet data on $ \partial \Omega $ under natural assumptions, and, when $ n = 1 $, we also show the existence of absolute minimisers. We further derive a necessary fully nonlinear PDE of third-order which arises as the analogue of the Euler-Lagrange equation for absolute minimisers, and is given by

$ \ \ \mathrm H_{\mathrm X}(\cdot,u,\mathrm D u,\mathrm D^2u): \mathrm D\big(\mathrm H(\cdot,u,\mathrm D u,\mathrm D^2u)\big)\otimes \mathrm D\big(\mathrm H(\cdot,u,\mathrm D u,\mathrm D^2u)\big) = 0\ \ \text{ in }\Omega. $

We then rigorously derive this PDE from smooth absolute minimisers, and prove the existence of generalised (merely measurable) solutions to the (first-order) Dirichlet problem on bounded domains. This generalises the key results obtained in [26] which first studied problems of this type, providing at the same time some simpler streamlined proofs.

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Mathematics Subject Classification: Primary 35J47, 35J60; Secondary 35D30, 35A15.

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References

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