Abstract
In this paper we consider the family of sets verifying the uniform cusp property introduced in [2] and extended in [4] to cusp functions only continuous at the origin. In the latter case we show that to any extended cusp function, we can associate a continuous, non-negative, and monotone strictly increasing cusp function of the type introduced in [2]. We construct an example of a bounded set in R N with a cusp function of the form c|θ|α, 0 < α < 1, for which its boundary integral is infinite and the Hausdorff dimension of its boundary is exactly N − α. We then give compactness theorems for the family of subsets of a bounded open holdall verifying a uniform cusp property with a uniform bound on either the De Georgi [6] or the γ-density perimeter of Bucur and Zolésio [1]. We also give their uniform local C 0-graph versions following [4]. This class forms a much larger family than the one of subsets verifying a uniform cone property.
This research has been supported by National Sciences and Engineering Research Council of Canada discovery grant A-8730 and by a FQRNT grant from the Ministère de l’Éducation du Québec.
Chapter PDF
Similar content being viewed by others
Keywords
References
D. Bucur and J.-P. Zolésio, Free boundary problems and density perimeter, J. Differential Equations 126 (1996), 224–243.
M.C. Delfour and J.-P. Zolésio, Shapes and Geometries: Analysis, Differential Calculus and Optimization, SIAM series on Advances in Design and Control, Society for Industrial and Applied Mathematics, Philadelphia, USA 2001.
M.C. Delfour and J.-P. Zolésio, The new family of cracked sets and the image segmentation problem revisited, CRM Report, May 2003, Université de Montréal, accepted in Communications in Information and Systems.
M.C. Delfour, N. Doyon, and J.-P. Zolésio, Extension of the uniform cusp property in shape optimization, in “Control of Partial Differential Equations”, G. Leugering, O. Imanuvilov, R. Triggiani, and B. Zhang, eds. Lectures Notes in Pure and Applied Mathematics, Marcel Dekker, May 2003, accepted.
M.C. Delfour, N. Doyon, and J.-P. Zolésio, The uniform fat segment and cusp properties in shape optimization, in “Control and Boundary analysis”, J. Cagnol and J.-P. Zolésio (Eds.), pp. 85–96, Marcel Dekker 2004.
E. Giusti, Minimal surfaces and functions of bounded variation, Birkhäuser, Boston, Basel, Stuttgart, 1984.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 International Federation for Information Processing
About this paper
Cite this paper
Delfour, M.C., Doyon, N., Zolésio, JP. (2005). Uniform Cusp Property, Boundary Integral, and Compactness for Shape Optimization. In: Cagnol, J., Zolésio, JP. (eds) System Modeling and Optimization. CSMO 2003. IFIP International Federation for Information Processing, vol 166. Springer, Boston, MA. https://doi.org/10.1007/0-387-23467-5_2
Download citation
DOI: https://doi.org/10.1007/0-387-23467-5_2
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4020-7760-9
Online ISBN: 978-0-387-23467-0
eBook Packages: Computer ScienceComputer Science (R0)