Explicit singular viscosity solutions of the Aronsson equation

Nikolaos I. Katzourakis

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

Nikolaos I. Katzourakis ¹

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

Comptes Rendus. Mathématique, Volume 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 :

Nikolaos I. Katzourakis 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},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1173--1176},
     publisher = {Elsevier},
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     number = {21-22},
     year = {2011},
     doi = {10.1016/j.crma.2011.10.010},
     language = {en},
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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/

Bibliographie
Cité par

[1] G. Aronsson On certain singular solutions of the partial differential equation ${(u_{x})}^{2} u_{x x} + 2 u_{x} u_{y} u_{x y} + {(u_{y})}^{2} u_{y y} = 0$ , Manuscripta Math., Volume 47 (1984) no. 1–3, pp. 133-151

[2] G. Aronsson Construction of singular solutions to the p-harmonic equation and its limit equation for $p = \infty$ , Manuscripta Math., Volume 56 (1986), pp. 135-158

[3] E.N. Barron; L.C. Evans; R. Jensen The infinity Laplacian, Aronssonʼs equation and their generalizations, Transactions of the AMS, Volume 360 (January 2008) no. 1 (electr. published on July 25, 2007)

[4] M.G. Crandall; H. Ishii; P.-L. Lions Userʼs guide to viscosity solutions of 2nd order partial differential equations, Bulletin of the AMS, Volume 27 (1992) no. 1, pp. 1-67

[5] T. De Pauw; A. Koeller Linearly approximatable functions, Proc. of the AMS (April 2009), pp. 1347-1356 (electr. published on October 6, 2008)

[6] L.C. Evans; O. Savin $C^{1, α}$ regularity for infinity harmonic functions in two dimensions, Calc. Var., Volume 32 (2008), pp. 325-347

[7] L.C. Evans, C.K. Smart, Everywhere differentiability of infinity harmonic functions, preprint.

[8] R. Gariepy; Ch. Wang; Y. Yu Generalized cone comparison principle for viscosity solutions of the Aronsson equation and absolute minimizers, Communications in PDE, Volume 31 (2006), pp. 1027-1046

[9] N.I. Katzourakis, A Hölder continuous nowhere improvable function with derivative singular distribution, preprint.

[10] O. Savin $C^{1}$ regularity for infinity harmonic functions in two dimensions, Arch. Rational Mech. Anal., Volume 176 (2005), pp. 351-361

[11] C. Wang; Y. Yu $C^{1}$ regularity of the Aronsson equation in $R^{2}$ , Ann. Inst. H. Poincaré, AN, Volume 25 (2008), pp. 659-678

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