Abstract
The average-distance problem, in the penalized formulation, involves minimizing $$ E\\mu^\\lambda(\\Sigma):=\\int{\\mathbb R^d} \\mathrm {inf}\_{y\\in\\Sigma} |x-y|\\mathrm d\\mu(x)+\\lambda\\mathcal H^1(\\Sigma), $$ among compact, connected sets $\\Sigma$, where $\\mathcal H^1$ denotes the 1-Hausdorff measure, $d\\geq 2$, $\\mu$ is a given measure and $\\lambda$ a given parameter. Regularity of minimizers is a delicate problem. It is known that even if $\\mu$ is absolutely continuous with respect to Lebesgue measure, $C^1$ regularity does not hold in general. An interesting question is whether the set of corners, i.e. points where $C^1$ regularity does not hold, is closed. The aim of this paper is to provide an example of minimizer whose set of corners is not closed, with reference measure $\\mu$ absolutely continuous with respect to Lebesgue measure.
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: Rendiconti del Seminario Matematico della Università di Padova
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.