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.

Abstract (VO)
Résumé

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.

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 $C^{1}$ .

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},
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     pages = {1173--1176},
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     volume = {349},
     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

Qianyun Miao; Fa Peng; Yuan Zhou A quantitative Sobolev regularity for absolute minimizers involving Hamiltonian $H (p)$ in plane, SIAM Journal on Mathematical Analysis, Volume 54 (2022) no. 6, pp. 5792-5853 | DOI:10.1137/21m1443601 | Zbl:1505.35170
Fa Peng; Changyou Wang; Yuan Zhou Regularity of absolute minimizers for continuous convex Hamiltonians, Journal of Differential Equations, Volume 274 (2021), pp. 1115-1164 | DOI:10.1016/j.jde.2020.11.011 | Zbl:1455.35035
Nikos Katzourakis Existence of Viscosity Solutions to the Dirichlet Problem for the $\infty$ -Laplacian, An Introduction To Viscosity Solutions for Fully Nonlinear PDE with Applications to Calculus of Variations in L∞ (2015), p. 101 | DOI:10.1007/978-3-319-12829-0_8
Nikos Katzourakis On the structure of $\infty$ -harmonic maps, Communications in Partial Differential Equations, Volume 39 (2014) no. 11, pp. 2091-2124 | DOI:10.1080/03605302.2014.920351 | Zbl:1304.49038
Nikolaos I. Katzourakis $\infty$ -minimal submanifolds, Proceedings of the American Mathematical Society, Volume 142 (2014) no. 8, pp. 2797-2811 | DOI:10.1090/s0002-9939-2014-12039-9 | Zbl:1301.35029
Nicholas Katzourakis Explicit 2D $\infty$ -harmonic maps whose interfaces have junctions and corners, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 351 (2013) no. 17-18, pp. 677-680 | DOI:10.1016/j.crma.2013.07.028 | Zbl:1278.35057
Nikolaos I. Katzourakis $L^{\infty}$ variational problems for maps and the Aronsson PDE system, Journal of Differential Equations, Volume 253 (2012) no. 7, pp. 2123-2139 | DOI:10.1016/j.jde.2012.05.012 | Zbl:1248.35074
Nikolaos I. Katzourakis A Hölder continuous nowhere improvable function with derivative singular distribution, S $\vec{e}$ MA Journal, Volume 59 (2012), pp. 37-51 | DOI:10.1007/bf03322609 | Zbl:1260.35051
Silika Prohl Solving Non-Linear PDE's for Mathematical Finance, SSRN Electronic Journal (2011) | DOI:10.2139/ssrn.2725441

Cité par 9 documents. Sources : Crossref, zbMATH

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