IDEAS home Printed from https://ideas.repec.org/a/hin/jijmms/269607.html
   My bibliography  Save this article

The Dirichlet Problem for the 2D Laplace Equation in a Domain with Cracks without Compatibility Conditions at Tips of the Cracks

Author

Listed:
  • P. A. Krutitskii

Abstract

We study the Dirichlet problem for the 2D Laplace equation in a domain bounded by smooth closed curves and smooth cracks. In the formulation of the problem, we do not require compatibility conditions for Dirichlet's boundary data at the tips of the cracks. However, if boundary data satisfies the compatibility conditions at the tips of the cracks, then this is a particular case of our problem. The cases of both interior and exterior domains are considered. The well-posed formulation of the problem is given, theorems on existence and uniqueness of a classical solution are proved, and the integral representation for a solution is obtained. It is shown that weak solution of the problem does not typically exist, though the classical solution exists. The asymptotic formulae for singularities of a solution gradient at the tips of the cracks are presented.

Suggested Citation

  • P. A. Krutitskii, 2012. "The Dirichlet Problem for the 2D Laplace Equation in a Domain with Cracks without Compatibility Conditions at Tips of the Cracks," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2012, pages 1-20, October.
    • Handle: RePEc:hin:jijmms:269607
    DOI: 10.1155/2012/269607
    as

    Download full text from publisher

    File URL: http://downloads.hindawi.com/journals/IJMMS/2012/269607.pdf
    Download Restriction: no
    File URL: http://downloads.hindawi.com/journals/IJMMS/2012/269607.xml
    Download Restriction: no
    File URL: https://libkey.io/10.1155/2012/269607?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
    ---><---

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. P. A. Krutitskii, 2013. "The Dirichlet Problem for the Equation in the Exterior of Nonclosed Lipschitz Surfaces," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2013, pages 1-4, February.

    More about this item

    Statistics

    Access and download statistics

    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:hin:jijmms:269607. 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: Mohamed Abdelhakeem (email available below). General contact details of provider: https://www.hindawi.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.