On Second-Order $L^\infty$ Variational Problems with Lower-Order Terms
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917890810183680 |
|---|---|
| author | Dutton, Ben Katzourakis, Nikos |
| author_facet | Dutton, Ben Katzourakis, Nikos |
| contents | In this paper we study $2$nd order $L^\infty$ variational problems, through seeking to minimise a supremal functional involving the Hessian of admissible functions as well as lower-order terms. Specifically, given a bounded domain $Ω\subseteq \mathbb R^n$ and $\mathrm H : Ω\times\big(\mathbb R \times\mathbb R^n \times \mathbb R^{n^{\otimes2}}_s \big) \to \mathbb R$, we consider the functional \[ \mathrm{E}_\infty(u, \mathcal{O}) :=\underset{ \mathcal{O}}{\mathrm{ess}\sup}\hspace{1mm}\mathrm H (\cdot,u,\mathrm D u,\mathrm D^2u ) , \ \ u\in W^{2,\infty}(Ω), \ \mathcal{O} \subseteq Ω\text{ measurable}. \] We establish the existence of minimisers subject to (first-order) Dirichlet data on $\partial Ω$ 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
$$ \ \ \mathrm H_{\mathrm X}(\cdot,u,\mathrm D u,\mathrm D^2u): \mathrm D\big(\mathrm H(\cdot,u,\mathrm D u,\mathrm D^2u)\big)\otimes \mathrm D\big(\mathrm H(\cdot,u,\mathrm D u,\mathrm D^2u)\big)=0\ \ \text{ in }Ω.
$$ We then rigorously derive this PDE from smooth absolute minimisers, and prove the existence of generalised D-solutions to the (first-order) Dirichlet problem. Our work generalises the key results obtained in [26] which first studied problems of this type with pure Hessian dependence only, providing at the same time considerably simpler streamlined proofs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_11701 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Second-Order $L^\infty$ Variational Problems with Lower-Order Terms Dutton, Ben Katzourakis, Nikos Analysis of PDEs 35J47, 35J60 (Primary) 35D30, 35A15 (Secondary) In this paper we study $2$nd order $L^\infty$ variational problems, through seeking to minimise a supremal functional involving the Hessian of admissible functions as well as lower-order terms. Specifically, given a bounded domain $Ω\subseteq \mathbb R^n$ and $\mathrm H : Ω\times\big(\mathbb R \times\mathbb R^n \times \mathbb R^{n^{\otimes2}}_s \big) \to \mathbb R$, we consider the functional \[ \mathrm{E}_\infty(u, \mathcal{O}) :=\underset{ \mathcal{O}}{\mathrm{ess}\sup}\hspace{1mm}\mathrm H (\cdot,u,\mathrm D u,\mathrm D^2u ) , \ \ u\in W^{2,\infty}(Ω), \ \mathcal{O} \subseteq Ω\text{ measurable}. \] We establish the existence of minimisers subject to (first-order) Dirichlet data on $\partial Ω$ 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 $$ \ \ \mathrm H_{\mathrm X}(\cdot,u,\mathrm D u,\mathrm D^2u): \mathrm D\big(\mathrm H(\cdot,u,\mathrm D u,\mathrm D^2u)\big)\otimes \mathrm D\big(\mathrm H(\cdot,u,\mathrm D u,\mathrm D^2u)\big)=0\ \ \text{ in }Ω. $$ We then rigorously derive this PDE from smooth absolute minimisers, and prove the existence of generalised D-solutions to the (first-order) Dirichlet problem. Our work generalises the key results obtained in [26] which first studied problems of this type with pure Hessian dependence only, providing at the same time considerably simpler streamlined proofs. |
| title | On Second-Order $L^\infty$ Variational Problems with Lower-Order Terms |
| topic | Analysis of PDEs 35J47, 35J60 (Primary) 35D30, 35A15 (Secondary) |
| url | https://arxiv.org/abs/2412.11701 |