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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: P. Cannarsa, P. Cardaliaguet and E. Giorgieri
Journal: Trans. Amer. Math. Soc. 359 (2007), 2741-2775
MSC (2000): Primary 58E10, 49N60, 26B35
Published electronically: January 25, 2007
MathSciNet review: 2286054
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Abstract: 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.


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Additional Information

P. Cannarsa
Affiliation: Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy
Email: cannarsa@axp.mat.uniroma2.it

P. Cardaliaguet
Affiliation: Université de Bretagne Occidentale, UFR des Sciences et Techniques, 6 Av. Le Gorgeu, BP 809, 29285 Brest, France
Email: Pierre.Cardaliaguet@univ-brest.fr

E. Giorgieri
Affiliation: Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy
Email: giorgier@axp.mat.uniroma2.it

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04259-6
PII: S 0002-9947(07)04259-6
Keywords: Normal distance, singularities, semiconcave functions, eikonal equation, viscosity solutions, H\"older continuous functions
Received by editor(s): April 4, 2005
Published electronically: January 25, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.