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

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     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},
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     year = {2018},
     doi = {10.1016/j.crma.2018.10.005},
     language = {en},
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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/

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Rafael López The translating soliton equation, Minimal surfaces: integrable systems and visualisation. M:iv workshops, 2016–19, Cham: Springer, 2021, pp. 187-216 | DOI:10.1007/978-3-030-68541-6_11 | Zbl:1483.35178
Cristian Enache; Rafael López Minimum principles and a priori estimates for some translating soliton type problems, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 187 (2019), pp. 352-364 | DOI:10.1016/j.na.2019.05.008 | Zbl:1427.35047

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