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Hölder regularity of the normal distance with an application to a PDE model for growing sandpiles.

Cannarsa, Piermarco; Giorgieri, Elena; Cardaliaguet, Pierre (2007), Hölder regularity of the normal distance with an application to a PDE model for growing sandpiles., Transactions of the American Mathematical Society, 359, 6, p. 2741-2775. http://dx.doi.org/10.1090/S0002-9947-07-04259-6

Type
Article accepté pour publication ou publié
Date
2007
Nom de la revue
Transactions of the American Mathematical Society
Volume
359
Numéro
6
Éditeur
American Mathematical Society
Pages
2741-2775
Identifiant publication
http://dx.doi.org/10.1090/S0002-9947-07-04259-6
Métadonnées
Afficher la notice complète
Auteur(s)
Cannarsa, Piermarco
Dipartimento di Matematica [Roma II] [DIPMAT]
Giorgieri, Elena
Dipartimento di Matematica [Roma II] [DIPMAT]
Cardaliaguet, Pierre
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Résumé (EN)
Given a bounded domain $ \Omega$ in $ \mathbb{R}^2$ with smooth boundary, the cut locus $ \overline \Sigma$ is the closure of the set of nondifferentiability points of the distance $ d$ from the boundary of $ \Omega$. The normal distance to the cut locus, $ \tau(x)$, is the map which measures the length of the line segment joining $ x$ to the cut locus along the normal direction $ Dd(x)$, whenever $ x\notin \overline \Sigma$. Recent results show that this map, restricted to boundary points, is Lipschitz continuous, as long as the boundary of $ \Omega$ is of class $ C^{2,1}$. Our main result is the global Hölder regularity of $ \tau$ in the case of a domain $ \Omega$ with analytic boundary. We will also show that the regularity obtained is optimal, as soon as the set of the so-called regular conjugate points is nonempty. In all the other cases, Lipschitz continuity can be extended to the whole domain $ \Omega$. The above regularity result for $ \tau$ is also applied to derive the Hölder continuity of the solution of a system of partial differential equations that arises in granular matter theory and optimal mass transfer.
Mots-clés
Normal distance; eikonal equation; singularities; semiconcave functions; Hölder continuous functions; viscosity solutions

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