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

П. А. Крутицкий | Zendy

Open Access

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

Author(s) -

П. А. Крутицкий

Publication year - 2012

Publication title -

international journal of mathematics and mathematical sciences

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.21

H-Index - 39

eISSN - 1687-0425

pISSN - 0161-1712

DOI - 10.1155/2012/269607

Subject(s) - mathematics , compatibility (geochemistry) , uniqueness , gravitational singularity , mathematical analysis , dirichlet problem , bounded function , laplace transform , boundary value problem , dirichlet distribution , laplace's equation , dirichlet boundary condition , geochemistry , geology

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

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