Hölder regularity of the normal distance with an application to a PDE model for growing sandpiles
HTML articles powered by AMS MathViewer
- by P. Cannarsa, P. Cardaliaguet and E. Giorgieri PDF
- Trans. Amer. Math. Soc. 359 (2007), 2741-2775 Request permission
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.References
- Paolo Albano and Piermarco Cannarsa, Propagation of singularities for solutions of nonlinear first order partial differential equations, Arch. Ration. Mech. Anal. 162 (2002), no. 1, 1–23. MR 1892229, DOI 10.1007/s002050100176
- L. Ambrosio, P. Cannarsa, and H. M. Soner, On the propagation of singularities of semi-convex functions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 20 (1993), no. 4, 597–616. MR 1267601
- Boutreux T., de Gennes P.-G., Surface flows of granular mixtures, I. General principles and minimal model. J. Phys. I France 6, (1996), 1295-1304.
- Piermarco Cannarsa and Pierre Cardaliaguet, Representation of equilibrium solutions to the table problem for growing sandpiles, J. Eur. Math. Soc. (JEMS) 6 (2004), no. 4, 435–464. MR 2094399, DOI 10.4171/JEMS/16
- P. Cannarsa, P. Cardaliaguet, G. Crasta, and E. Giorgieri, A boundary value problem for a PDE model in mass transfer theory: representation of solutions and applications, Calc. Var. Partial Differential Equations 24 (2005), no. 4, 431–457. MR 2180861, DOI 10.1007/s00526-005-0328-7
- Hyeong In Choi, Sung Woo Choi, and Hwan Pyo Moon, Mathematical theory of medial axis transform, Pacific J. Math. 181 (1997), no. 1, 57–88. MR 1491036, DOI 10.2140/pjm.1997.181.57
- F. H. Clarke, Yu. S. Ledyaev, R. J. Stern, and P. R. Wolenski, Nonsmooth analysis and control theory, Graduate Texts in Mathematics, vol. 178, Springer-Verlag, New York, 1998. MR 1488695
- L. C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc. 137 (1999), no. 653, viii+66. MR 1464149, DOI 10.1090/memo/0653
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
- Hadeler K.P., Kuttler C., Dynamical models for granular matter, Granular Matter 2, (1999), 9-18.
- Jin-ichi Itoh and Minoru Tanaka, The Lipschitz continuity of the distance function to the cut locus, Trans. Amer. Math. Soc. 353 (2001), no. 1, 21–40. MR 1695025, DOI 10.1090/S0002-9947-00-02564-2
- Yanyan Li and Louis Nirenberg, The distance function to the boundary, Finsler geometry, and the singular set of viscosity solutions of some Hamilton-Jacobi equations, Comm. Pure Appl. Math. 58 (2005), no. 1, 85–146. MR 2094267, DOI 10.1002/cpa.20051
- Pierre-Louis Lions, Generalized solutions of Hamilton-Jacobi equations, Research Notes in Mathematics, vol. 69, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. MR 667669
- Carlo Mantegazza and Andrea Carlo Mennucci, Hamilton-Jacobi equations and distance functions on Riemannian manifolds, Appl. Math. Optim. 47 (2003), no. 1, 1–25. MR 1941909, DOI 10.1007/s00245-002-0736-4
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
- MR Author ID: 323521
- 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
- Received by editor(s): April 4, 2005
- Published electronically: January 25, 2007
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 359 (2007), 2741-2775
- MSC (2000): Primary 58E10, 49N60, 26B35
- DOI: https://doi.org/10.1090/S0002-9947-07-04259-6
- MathSciNet review: 2286054