Comptes Rendus
Partial Differential Equations
Explicit singular viscosity solutions of the Aronsson equation
Comptes Rendus. Mathématique, Volume 349 (2011) no. 21-22, pp. 1173-1176.

We establish that when n2 and HC1(Rn) is a Hamiltonian such that some level set contains a line segment, the Aronsson equation D2u:Hp(Du)Hp(Du)=0 admits explicit entire viscosity solutions. They are superpositions of a linear part plus a Lipschitz continuous singular part which in general is non-C1 and nowhere twice differentiable. In particular, we supplement the C1 regularity result of Wang and Yu (2008) [11] by deducing that strict level convexity is necessary for C1 regularity of solutions.

Nous démontrons que, pour n2 et un Hamiltonien HC1(Rn) tel quʼau moins une de ses lignes de niveau contienne un segment de droite, lʼéquation de Aronsson D2u:Hp(Du)Hp(Du)=0 admet des solutions de viscosité explicites définies sur Rn. Elles sont superpositions dʼune partie linéaire et dʼune partie continue, lipschitzienne, singulière qui, en général, nʼest pas C1 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 C1.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2011.10.010

Nikolaos I. Katzourakis 1

1 BCAM – Basque Center for Applied Mathematics, Biskaia Technology Park, Building 500, 48160 Derio, Spain
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Nikolaos I. Katzourakis. Explicit singular viscosity solutions of the Aronsson equation. Comptes Rendus. Mathématique, Volume 349 (2011) no. 21-22, pp. 1173-1176. doi : 10.1016/j.crma.2011.10.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.10.010/

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