On the Square Distance Function from a Manifold with Boundary

Giovanni Bellettini; Alaa Elshorbagy

doi:10.68381/jca29025

Abstract

We characterize arbitrary codimensional smooth manifolds

\mathcal{M}

with boundary embedded in

\mathbb{R}^n

using the square distance function from

\mathcal{M}

and from its boundary. The results are localized in an open set.

Author information

Contact details are reproduced from the original publication and may be historical.

Giovanni Bellettini

Dip. di Ingegneria dell'Informazione e Scienze Matematiche, Università di Siena, Italy
and: International Centre for Theoretical Physics, Mathematics Section, Trieste, Italy

bellettini@diism.unisi.it

Alaa Elshorbagy

Area of Mathematical Analysis, Modelling and Applications, SISSA, Trieste, Italy
and: International Centre for Theoretical Physics, Mathematics Section, Trieste, Italy

alaa.elshorbagy@sissa.it

Keywords

Square distance function
smooth manifolds with boundary
signed distance function

Mathematics Subject Classification

53A07; 51K05

References

L. Ambrosio: Lecture notes on geometric evolution problems, distance function and viscosity solutions, in: Calculus of Variations and Partial Differential Equations. Topics on Geometrical Evolution Problems and Degree Theory, Springer, Berlin (1999) 5–94.https://doi.org/10.1007/978-3-642-57186-2_2 Google Scholar
L. Ambrosio, C. Mantegazza: Curvature and distance function from a manifold, J. Geom. Analysis 8 (1998) 723–748.https://doi.org/10.1007/BF02922668 Google Scholar
L. Ambrosio, H. M. Soner: Level set approach to mean curvature flow in arbitrary codimension, J. Diff. Geom. 43 (1996) 693–737.https://doi.org/10.4310/jdg/1214458529 Google Scholar
G. Bellettini, M. Masala, M. Novaga: On a conjecture of De Giorgi on the squared distance function, J. Convex Analysis 14 (2007) 353–359.https://doi.org/10.68381/jca14024 Google Scholar
G. Bellettini, M. Novaga, M. Paolini: An example of three-dimensional fattening for linked space curves evolving by curvature, Comm. Partial Diff. Equations 23 (1998) 1475–1492.https://doi.org/10.1080/03605309808821392 Google Scholar
P. Cannarsa, C. Sinestrari: Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Progress in Nonlinear Differential Equations and Their Applications 58, Boston, Birkhäuser (2004).https://doi.org/10.1007/b138356 Google Scholar
E. De Giorgi: Introduzione ai problemi di discontinuità libera, Symmetry in Nature I, Scuola Norm. Sup. Pisa (1989) 265–285.Google Scholar
E. De Giorgi: Barriers, boundaries, motion of manifolds, Conference held at the Department of Mathematics of Pavia, March 18 (1994).Google Scholar
M. C. Delfour, J.-P. Zolésio: Shapes and Geometries. Analysis, Differential Calculus, and Optimization, Advances in Design and Control 4, SIAM, Philadelphia (2011).https://doi.org/10.1137/1.9780898719826 Google Scholar
L. C. Evans, J. Spruck: Motion of level sets by mean curvature II, Trans. Amer. Math. Soc. 330 (1992) 321–332.https://doi.org/10.1090/S0002-9947-1992-1068927-8 Google Scholar
H. Federer: Curvature measures, Trans. Amer. Math. Soc. 93 (1959) 418–491.https://doi.org/10.1090/S0002-9947-1959-0110078-1 Google Scholar
D. Gilbarg, N. S. Trudinger: Elliptic Partial Differential Equations of Second Order, Springer, Berlin (2001).https://doi.org/10.1007/978-3-642-61798-0 Google Scholar
E. Giusti: Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Boston (1984).https://doi.org/10.1007/978-1-4684-9486-0 Google Scholar
V. Guillemin, A. Pollack: Differential Topology, Prentice-Hall, Englewood Cliffs (1974).Google Scholar
J.-B. Poly and G. Raby: Fonctions distance et singularités, Bull. Sci. Math. 108 (1984) 187–195.Google Scholar
D. Slepcev: On level set approach to motion of manifolds of arbitrary codimension, Interfaces Free Bound. 5 (2003) 417–458.https://doi.org/10.4171/ifb/86 Google Scholar
M. Spivak: Calculus on Manifolds, Westview Press, Boulder (1965).Google Scholar

Suggested citation

Published by Heldermann Verlag, 2022. Rights now held by Banach Press LLC.

[1] L. Ambrosio: Lecture notes on geometric evolution problems, distance function and viscosity solutions, in: Calculus of Variations and Partial Differential Equations. Topics on Geometrical Evolution Problems and Degree Theory, Springer, Berlin (1999) 5–94.https://doi.org/10.1007/978-3-642-57186-2_2 Google Scholar

[2] L. Ambrosio, C. Mantegazza: Curvature and distance function from a manifold, J. Geom. Analysis 8 (1998) 723–748.https://doi.org/10.1007/BF02922668 Google Scholar

[3] L. Ambrosio, H. M. Soner: Level set approach to mean curvature flow in arbitrary codimension, J. Diff. Geom. 43 (1996) 693–737.https://doi.org/10.4310/jdg/1214458529 Google Scholar

[4] G. Bellettini, M. Masala, M. Novaga: On a conjecture of De Giorgi on the squared distance function, J. Convex Analysis 14 (2007) 353–359.https://doi.org/10.68381/jca14024 Google Scholar

[5] G. Bellettini, M. Novaga, M. Paolini: An example of three-dimensional fattening for linked space curves evolving by curvature, Comm. Partial Diff. Equations 23 (1998) 1475–1492.https://doi.org/10.1080/03605309808821392 Google Scholar

[6] P. Cannarsa, C. Sinestrari: Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Progress in Nonlinear Differential Equations and Their Applications 58, Boston, Birkhäuser (2004).https://doi.org/10.1007/b138356 Google Scholar

[7] E. De Giorgi: Introduzione ai problemi di discontinuità libera, Symmetry in Nature I, Scuola Norm. Sup. Pisa (1989) 265–285.Google Scholar

[8] E. De Giorgi: Barriers, boundaries, motion of manifolds, Conference held at the Department of Mathematics of Pavia, March 18 (1994).Google Scholar

[9] M. C. Delfour, J.-P. Zolésio: Shapes and Geometries. Analysis, Differential Calculus, and Optimization, Advances in Design and Control 4, SIAM, Philadelphia (2011).https://doi.org/10.1137/1.9780898719826 Google Scholar

[10] L. C. Evans, J. Spruck: Motion of level sets by mean curvature II, Trans. Amer. Math. Soc. 330 (1992) 321–332.https://doi.org/10.1090/S0002-9947-1992-1068927-8 Google Scholar

[11] H. Federer: Curvature measures, Trans. Amer. Math. Soc. 93 (1959) 418–491.https://doi.org/10.1090/S0002-9947-1959-0110078-1 Google Scholar

[12] D. Gilbarg, N. S. Trudinger: Elliptic Partial Differential Equations of Second Order, Springer, Berlin (2001).https://doi.org/10.1007/978-3-642-61798-0 Google Scholar

[13] E. Giusti: Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Boston (1984).https://doi.org/10.1007/978-1-4684-9486-0 Google Scholar

[14] V. Guillemin, A. Pollack: Differential Topology, Prentice-Hall, Englewood Cliffs (1974).Google Scholar

[15] J.-B. Poly and G. Raby: Fonctions distance et singularités, Bull. Sci. Math. 108 (1984) 187–195.Google Scholar

[16] D. Slepcev: On level set approach to motion of manifolds of arbitrary codimension, Interfaces Free Bound. 5 (2003) 417–458.https://doi.org/10.4171/ifb/86 Google Scholar

[17] M. Spivak: Calculus on Manifolds, Westview Press, Boulder (1965).Google Scholar