Explicit singular viscosity solutions of the Aronsson equation

Katzourakis, Nikolaos I.

doi:10.1016/j.crma.2011.10.010

Partial Differential Equations

Explicit singular viscosity solutions of the Aronsson equation
[Solutions de viscosité explicites de lʼéquation de Aronsson]

Présenté par : the Editorial Board

Katzourakis, Nikolaos I.¹

¹ BCAM – Basque Center for Applied Mathematics, Biskaia Technology Park, Building 500, 48160 Derio, Spain

Comptes Rendus. Mathématique, Tome 349 (2011) no. 21-22, pp. 1173-1176.

Résumé
Abstract

Nous démontrons que, pour $n ⩾ 2$ et un Hamiltonien $H \in C^{1} (R^{n})$ tel quʼau moins une de ses lignes de niveau contienne un segment de droite, lʼéquation de Aronsson $D^{2} u : H_{p} (D u) \otimes H_{p} (D u) = 0$ admet des solutions de viscosité explicites définies sur $R^{n}$ . Elles sont superpositions dʼune partie linéaire et dʼune partie continue, lipschitzienne, singulière qui, en général, nʼest pas $C^{1}$ et est nulle part deux fois dérivable. Plus précisément, nous complétons le résultat de régularité établit par Wang et Yu (2008) [11] en montrant que la stricte convexité des lignes de niveau est nécessaire pour que les solutions soient $C^{1}$ .

We establish that when $n ⩾ 2$ and $H \in C^{1} (R^{n})$ is a Hamiltonian such that some level set contains a line segment, the Aronsson equation $D^{2} u : H_{p} (D u) \otimes H_{p} (D u) = 0$ admits explicit entire viscosity solutions. They are superpositions of a linear part plus a Lipschitz continuous singular part which in general is non- $C^{1}$ and nowhere twice differentiable. In particular, we supplement the $C^{1}$ regularity result of Wang and Yu (2008) [11] by deducing that strict level convexity is necessary for $C^{1}$ regularity of solutions.

Reçu le : 2011-09-01
Accepté le : 2011-10-12
Publié le : 2011-10-28

DOI : 10.1016/j.crma.2011.10.010

Affiliations des auteurs :

Katzourakis, Nikolaos I. ¹

¹ BCAM – Basque Center for Applied Mathematics, Biskaia Technology Park, Building 500, 48160 Derio, Spain

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     title = {Explicit singular viscosity solutions of the {Aronsson} equation},
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Katzourakis, Nikolaos I. Explicit singular viscosity solutions of the Aronsson equation. Comptes Rendus. Mathématique, Tome 349 (2011) no. 21-22, pp. 1173-1176. doi : 10.1016/j.crma.2011.10.010. http://www.numdam.org/articles/10.1016/j.crma.2011.10.010/

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Cité par

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