The Dirichlet problem on a strip for the <i>α</i>-translating soliton equation

Rafael López

doi:10.1016/j.crma.2018.10.005

Partial differential equations/Differential geometry

The Dirichlet problem on a strip for the α-translating soliton equation
[Le problème de Dirichlet pour l'équation α-soliton de translation dans une bande]

Présenté par : Haïm Brézis

Rafael López ¹

¹ Departamento de Geometría y Topología, Instituto de Matemáticas (IEMath-GR), Universidad de Granada, 18071 Granada, Spain

Comptes Rendus. Mathématique, Volume 356 (2018) no. 11-12, pp. 1179-1187.

Résumé
Abstract

Dans cette note, nous prouvons l'existence de solutions classiques au problème de Dirichlet pour l'équation de α-soliton de translation définie dans une bande de $R^{2}$ ; les données sur le bord sont deux copies d'une fonction convexe continue. Nous utilisons la méthode de Perron, dans laquelle une famille de grim reapers est employée comme barrière pour résoudre le problème de Dirichlet.

In this paper, we investigate the Dirichlet problem associated with the α-translating equation. Using the Perron method and a family of grim reapers as barriers, we prove the existence of a solution on a strip of $R^{2}$ and the boundary data is formed by two copies of a convex function.

Reçu le : 2018-04-17
Accepté le : 2018-10-22
Publié le : 2018-10-29

DOI : 10.1016/j.crma.2018.10.005

Affiliations des auteurs :

Rafael López ¹

¹ Departamento de Geometría y Topología, Instituto de Matemáticas (IEMath-GR), Universidad de Granada, 18071 Granada, Spain

@article{CRMATH_2018__356_11-12_1179_0,
     author = {Rafael L\'opez},
     title = {The {Dirichlet} problem on a strip for the \protect\emph{\ensuremath{\alpha}}-translating soliton equation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1179--1187},
     publisher = {Elsevier},
     volume = {356},
     number = {11-12},
     year = {2018},
     doi = {10.1016/j.crma.2018.10.005},
     language = {en},
}

TY  - JOUR
AU  - Rafael López
TI  - The Dirichlet problem on a strip for the α-translating soliton equation
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 1179
EP  - 1187
VL  - 356
IS  - 11-12
PB  - Elsevier
DO  - 10.1016/j.crma.2018.10.005
LA  - en
ID  - CRMATH_2018__356_11-12_1179_0
ER  -

%0 Journal Article
%A Rafael López
%T The Dirichlet problem on a strip for the α-translating soliton equation
%J Comptes Rendus. Mathématique
%D 2018
%P 1179-1187
%V 356
%N 11-12
%I Elsevier
%R 10.1016/j.crma.2018.10.005
%G en
%F CRMATH_2018__356_11-12_1179_0

Rafael López. The Dirichlet problem on a strip for the α-translating soliton equation. Comptes Rendus. Mathématique, Volume 356 (2018) no. 11-12, pp. 1179-1187. doi : 10.1016/j.crma.2018.10.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.10.005/

Bibliographie
Cité par

[1] S.J. Altschuler; L.F. Wu Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle, Calc. Var. Partial Differ. Equ., Volume 2 (1994), pp. 101-111

[2] M. Bergner The Dirichlet problem for graphs of prescribed anisotropic mean curvature in $R^{n + 1}$ , Analysis, Volume 28 (2008), pp. 149-166

[3] S. Bernstein Sur les surfaces définies au moyen de leur courbure moyenne ou totale, Ann. Sci. Éc. Norm. Supér., Volume 27 (2010), pp. 233-256

[4] J. Clutterbuck; O. Schnürer; F. Schulze Stability of translating solutions to mean curvature flow, Calc. Var. Partial Differ. Equ., Volume 29 (2007), pp. 281-293

[5] P. Collin Deux exemples de graphes de courbure moyenne constante sur une band de $R^{2}$ , C. R. Acad. Sci. Paris, Ser. I, Volume 311 (1990), pp. 539-542

[6] P. Collin; R. Krust Le problème de Dirichlet pour l'équation des surfaces minimales sur des domaines non bornés, Bull. Soc. Math. Fr., Volume 119 (1991), pp. 443-462

[7] R. Courant; D. Hilbert Methods of Mathematical Physics, Vol. II, Interscience, New York, 1962

[8] D. Gilbarg; N.S. Trudinger Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983

[9] C. Gui; H-Y. Jian; H-J. Ju Properties of translating solutions to mean curvature flow, Discrete Contin. Dyn. Syst., Volume 28B (2010), pp. 441-453

[10] G. Huisken; C. Sinestrari Mean curvature flow singularities for mean convex surfaces, Calc. Var., Volume 8 (1999), pp. 1-14

[11] T. Ilmanen Elliptic regularization and partial regularity for motion by mean curvature, Mem. Am. Math. Soc., Volume 108 (1994) no. 520 (x+90 pp)

[12] H-Y. Jian; H-J. Ju Existence of translating solutions to the flow by powers of mean curvature on unbounded domains, J. Differ. Equ., Volume 250 (2011), pp. 3967-3987

[13] H-J. Ju; Y. Liu Dirichlet problem for anisotropic prescribed mean curvature equation on unbounded domains, J. Math. Anal. Appl., Volume 439 (2016), pp. 709-724

[14] R. López Constant mean curvature graphs on unbounded convex domains, J. Differ. Equ., Volume 171 (2001), pp. 54-62

[15] T. Marquardt Remark on the anisotropic prescribed mean curvature equation on arbitrary domain, Math. Z., Volume 264 (2010), pp. 507-511

[16] F. Schulze Evolution of convex hypersurfaces by powers of the mean curvature, Math. Z., Volume 251 (2005), pp. 721-733

[17] F. Schulze Nonlinear evolution by mean curvature and isoperimetric inequalities, J. Differ. Geom., Volume 79 (2008), pp. 197-241

[18] J. Serrin The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables, Philos. Trans. R. Soc. Lond., Volume 264 (1969), pp. 413-496

[19] W. Sheng; C. Wu On asymptotic behavior for singularities of the powers of mean curvature flow, Chin. Ann. Math., Ser. B, Volume 30 (2009), pp. 51-66

[20] W. Sheng; C. Wu Rotationally symmetric translating soliton of $H^{k}$ -flow, Sci. China Math., Volume 53 (2010), pp. 1011-1016

[21] L. Simon Equations of mean curvature type in 2 independent variable, Pac. J. Math., Volume 69 (1977), pp. 245-268

[22] J. Spruck; L. Xiao Complete translating solitons to the mean curvature flow in $R^{3}$ with nonnegative mean curvature (preprint) | arXiv

[23] X-J. Wang Convex solutions to the mean curvature flow, Ann. of Math. (2), Volume 173 (2011), pp. 1185-1239

[24] B. White The nature of singularities in mean curvature flow of mean convex surfaces, J. Am. Math. Soc., Volume 16 (2003), pp. 123-138

Cité par Sources :

Commentaires - Politique