The Dirichlet Problem for the 2D Laplace Equation in a Domain with Cracks without Compatibility Conditions at Tips of the Cracks
Author
Source
International Journal of Mathematics and Mathematical Sciences
Issue
Vol. 2012, Issue 2012 (31 Dec. 2012), pp.1-20, 20 p.
Publisher
Hindawi Publishing Corporation
Publication Date
2012-10-31
Country of Publication
Egypt
No. of Pages
20
Main Subjects
Abstract EN
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.
American Psychological Association (APA)
Krutitskii, P. A.. 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،Vol. 2012, no. 2012, pp.1-20.
https://search.emarefa.net/detail/BIM-459047
Modern Language Association (MLA)
Krutitskii, P. A.. 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 No. 2012 (2012), pp.1-20.
https://search.emarefa.net/detail/BIM-459047
American Medical Association (AMA)
Krutitskii, P. A.. 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. 2012. Vol. 2012, no. 2012, pp.1-20.
https://search.emarefa.net/detail/BIM-459047
Data Type
Journal Articles
Language
English
Notes
Includes bibliographical references
Record ID
BIM-459047