We establish that when and is a Hamiltonian such that some level set contains a line segment, the Aronsson equation admits explicit entire viscosity solutions. They are superpositions of a linear part plus a Lipschitz continuous singular part which in general is non- and nowhere twice differentiable. In particular, we supplement the regularity result of Wang and Yu (2008) [11] by deducing that strict level convexity is necessary for regularity of solutions.
Nous démontrons que, pour et un Hamiltonien tel quʼau moins une de ses lignes de niveau contienne un segment de droite, lʼéquation de Aronsson admet des solutions de viscosité explicites définies sur . Elles sont superpositions dʼune partie linéaire et dʼune partie continue, lipschitzienne, singulière qui, en général, nʼest pas 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 .
Accepted:
Published online:
Nikolaos I. Katzourakis 1
@article{CRMATH_2011__349_21-22_1173_0, author = {Nikolaos I. Katzourakis}, title = {Explicit singular viscosity solutions of the {Aronsson} equation}, journal = {Comptes Rendus. Math\'ematique}, pages = {1173--1176}, publisher = {Elsevier}, volume = {349}, number = {21-22}, year = {2011}, doi = {10.1016/j.crma.2011.10.010}, language = {en}, }
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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