IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v182y2019i3d10.1007_s10957-019-01520-z.html
   My bibliography  Save this article

Approximation of Lipschitz Functions Preserving Boundary Values

Author

Listed:
  • Robert Deville

    (Université de Bordeaux 1)

  • Carlos Mudarra

    (Instituto de Ciencias Matemáticas (CSIC-UAM-UC3-UCM))

Abstract

Given an open subset of a Banach space and a Lipschitz real-valued function defined on its closure, we study whether it is possible to approximate this function uniformly by Lipschitz functions having the same Lipschitz constant and preserving the values of the initial function on the boundary of the open set, and which are k times continuously differentiable on the open. A consequence of our result is that every 1-Lipschitz function defined on the closure of an open subset of a finite-dimensional normed space of dimension greater than one, and such that the Lipschitz constant of its restriction to the boundary is less than 1, can be uniformly approximated by differentiable 1-Lipschitz functions preserving the values of the initial function on the boundary of the open set, and such that its derivative has norm one almost everywhere on the open. This result does not hold in general without assumption on the Lipschitz constant of the restriction of the initial function to the boundary.

Suggested Citation

  • Robert Deville & Carlos Mudarra, 2019. "Approximation of Lipschitz Functions Preserving Boundary Values," Journal of Optimization Theory and Applications, Springer, vol. 182(3), pages 885-905, September.
    • Handle: RePEc:spr:joptap:v:182:y:2019:i:3:d:10.1007_s10957-019-01520-z
    DOI: 10.1007/s10957-019-01520-z
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-019-01520-z
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-019-01520-z?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:joptap:v:182:y:2019:i:3:d:10.1007_s10957-019-01520-z. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.