Abstract
We study the Riemannian distance function from a fixed point (a point-wise target) of Euclidean space in the presence of a compact obstacle bounded by a smooth hypersurface. First, we show that such a function is locally semiconcave with a fractional modulus of order one half and that, near the obstacle, this regularity is optimal. Then, in the Euclidean setting, we prove that the singularities of the distance function propagate, in the sense that each singular point belongs to a nontrivial singular continuum. Finally, we investigate the lack of differentiability of the distance function when a convex obstacle is present.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
More From: Calculus of Variations and Partial Differential Equations
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.