Solutions of vectorial Hamilton–Jacobi equations are rank-one absolute minimisers in L∞L^{\infty}
Given the supremal functional E∞(u,Ω′)=esssupΩ′H(⋅,Du){E_{\infty}(u,\Omega^{\prime})=\operatornamewithlimits{ess\,sup}_{\Omega^{% \prime}}H(\,\cdot\,,\mathrm{D}u)}, defined on Wloc1,∞(Ω,ℝN){W^{1,\infty}_{\mathrm{loc}}(\Omega,\mathbb{R}^{N})}, with Ω′⋐Ω⊆ℝn{\Omega^{\prime}\Subset\Omega\subseteq\...
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| Format: | Article |
| Language: | English |
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De Gruyter
2017-06-01
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| Series: | Advances in Nonlinear Analysis |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/anona-2016-0164 |
| _version_ | 1819290869136621568 |
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| author | Katzourakis Nikos |
| author_facet | Katzourakis Nikos |
| author_sort | Katzourakis Nikos |
| collection | DOAJ |
| description | Given the supremal functional E∞(u,Ω′)=esssupΩ′H(⋅,Du){E_{\infty}(u,\Omega^{\prime})=\operatornamewithlimits{ess\,sup}_{\Omega^{%
\prime}}H(\,\cdot\,,\mathrm{D}u)}, defined on Wloc1,∞(Ω,ℝN){W^{1,\infty}_{\mathrm{loc}}(\Omega,\mathbb{R}^{N})}, with Ω′⋐Ω⊆ℝn{\Omega^{\prime}\Subset\Omega\subseteq\mathbb{R}^{n}}, we identify a class of vectorial rank-one absolute minimisers by proving a statement slightly stronger than the next claim: vectorial solutions of the Hamilton–Jacobi equation H(⋅,Du)=c{H(\,\cdot\,,\mathrm{D}u)=c} are rank-one absolute minimisers if they are C1{C^{1}}.
Our minimality notion is a generalisation of the classical L∞{L^{\infty}} variational principle of Aronsson to the vector case, and emerged in earlier work of the author. The assumptions are minimal, requiring only continuity and rank-one convexity of the level sets. |
| first_indexed | 2024-12-24T03:29:35Z |
| format | Article |
| id | doaj.art-a6aa0ccd89e8475c9e7f4090f3672821 |
| institution | Directory Open Access Journal |
| issn | 2191-9496 2191-950X |
| language | English |
| last_indexed | 2024-12-24T03:29:35Z |
| publishDate | 2017-06-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Advances in Nonlinear Analysis |
| spelling | doaj.art-a6aa0ccd89e8475c9e7f4090f36728212022-12-21T17:17:15ZengDe GruyterAdvances in Nonlinear Analysis2191-94962191-950X2017-06-018150851610.1515/anona-2016-0164anona-2016-0164Solutions of vectorial Hamilton–Jacobi equations are rank-one absolute minimisers in L∞L^{\infty}Katzourakis Nikos0Department of Mathematics and Statistics, University of Reading, Whiteknights, PO Box 220, ReadingRG6 6AX, Berkshire, United KingdomGiven the supremal functional E∞(u,Ω′)=esssupΩ′H(⋅,Du){E_{\infty}(u,\Omega^{\prime})=\operatornamewithlimits{ess\,sup}_{\Omega^{% \prime}}H(\,\cdot\,,\mathrm{D}u)}, defined on Wloc1,∞(Ω,ℝN){W^{1,\infty}_{\mathrm{loc}}(\Omega,\mathbb{R}^{N})}, with Ω′⋐Ω⊆ℝn{\Omega^{\prime}\Subset\Omega\subseteq\mathbb{R}^{n}}, we identify a class of vectorial rank-one absolute minimisers by proving a statement slightly stronger than the next claim: vectorial solutions of the Hamilton–Jacobi equation H(⋅,Du)=c{H(\,\cdot\,,\mathrm{D}u)=c} are rank-one absolute minimisers if they are C1{C^{1}}. Our minimality notion is a generalisation of the classical L∞{L^{\infty}} variational principle of Aronsson to the vector case, and emerged in earlier work of the author. The assumptions are minimal, requiring only continuity and rank-one convexity of the level sets.https://doi.org/10.1515/anona-2016-0164rank-one absolute minimisersvectorial hamilton–jacobi equationviscosity solutions.35d99 35d40 35j47 35j92 35j70 35j99 |
| spellingShingle | Katzourakis Nikos Solutions of vectorial Hamilton–Jacobi equations are rank-one absolute minimisers in L∞L^{\infty} Advances in Nonlinear Analysis rank-one absolute minimisers vectorial hamilton–jacobi equation viscosity solutions. 35d99 35d40 35j47 35j92 35j70 35j99 |
| title | Solutions of vectorial Hamilton–Jacobi equations are rank-one absolute minimisers in L∞L^{\infty} |
| title_full | Solutions of vectorial Hamilton–Jacobi equations are rank-one absolute minimisers in L∞L^{\infty} |
| title_fullStr | Solutions of vectorial Hamilton–Jacobi equations are rank-one absolute minimisers in L∞L^{\infty} |
| title_full_unstemmed | Solutions of vectorial Hamilton–Jacobi equations are rank-one absolute minimisers in L∞L^{\infty} |
| title_short | Solutions of vectorial Hamilton–Jacobi equations are rank-one absolute minimisers in L∞L^{\infty} |
| title_sort | solutions of vectorial hamilton jacobi equations are rank one absolute minimisers in l∞l infty |
| topic | rank-one absolute minimisers vectorial hamilton–jacobi equation viscosity solutions. 35d99 35d40 35j47 35j92 35j70 35j99 |
| url | https://doi.org/10.1515/anona-2016-0164 |
| work_keys_str_mv | AT katzourakisnikos solutionsofvectorialhamiltonjacobiequationsarerankoneabsoluteminimisersinllinfty |